tag:blogger.com,1999:blog-221497182014-10-06T16:47:19.449-07:00Fundamentals of Piano PracticeRosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.comBlogger137125tag:blogger.com,1999:blog-22149718.post-1166390033934413542006-12-17T13:13:00.000-08:002006-12-17T13:13:53.976-08:00Content of Table<div align="center"><strong>Chapter 1</strong></div><div align="center"><strong></strong></div><div align="center"></div><div align="center"><strong></strong></div><div align="center"><strong></strong></div><div align="center"><span style="font-size:180%;"><strong>Technique</strong></span>
</div><div align="left">
<span style="font-size:130%;"><strong>Introduction
</strong></span></div><ul><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/objective.html">Objective</a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/what-is-piano-technique.html">What is Piano Technique?</a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/technique-and-music.html">Technique and Music</a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/basic-approach-interpretation-musical.html">Basic Approach, Interpretation, Musical Training, Perfect Pitch</a></li></ul><p><strong><span style="font-size:130%;">Basic Procedures for Piano Practice</span></strong></p><ul><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/practice-routine.html">The Practice Routine </a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/finger-positions.html">Finger Positions</a> </li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/bench-height-and-distance-from-piano.html">Bench Height and Distance from Piano </a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/starting-piece-listening-and-analysis.html">Starting a Piece: Listening and Analysis (Fur Elise)</a> </li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/practice-most-difficult-sections-first.html">Practice the Most Difficult Sections First </a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/shortening-difficult-passages.html">Shortening Difficult Passages: Segmental (Bar-by-Bar) Practice </a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/hands-separate-practice-acquiring.html">Hands Separate Practice: Acquiring Technique </a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/continuity-rule.html">The Continuity Rule</a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/chord-attack.html">The Chord Attack</a> </li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/gravity-drop-chord-practice-and.html">Gravity Drop, Chord Practice, and Relaxation </a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/parallel-sets.html">Parallel Sets</a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/learning-and-memorizing.html">Learning and Memorizing </a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/velocity-choice-of-practice-speed.html">Velocity, Choice of Practice Speed </a></li><li><a href="http://fundamentalpiano1.blogspot.com/2006/03/how-to-relax.html">How to Relax </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/post-practice-improvement-ppi.html">Post Practice Improvement (PPI)</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/dangers-of-slow-play-pitfalls-of.html">Dangers of Slow Play - Pitfalls of the Intuitive Method </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/importance-of-slow-play.html">Importance of Slow Play </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/fingering.html">Fingering</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/accurate-tempo-and-metronome.html">Accurate Tempo and the Metronome </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/weak-left-hand-using-one-hand-to-teach.html">Weak Left Hand; Using One Hand to Teach the Other</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/building-endurance-breathing.html">Building Endurance, Breathing</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/bad-habits-pianists-worst-enemy.html">Bad Habits: A Pianist's Worst Enemy </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/damper-pedal.html">Damper Pedal </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/soft-pedal-timbre-and-normal-modes-of.html">Soft Pedal, Timbre, and Normal Modes of Vibrating Strings </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/hands-together-chopins-fantaisie.html">Hands Together: Chopin's Fantaisie Impromptu </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/summary.html">Summary</a> </li></ul><p><strong><span style="font-size:130%;">Selected Topics in Piano Practice</span></strong></p><p>1. Tone, Rhythm and Staccato</p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/what-is-good-tone.html">What is "Good Tone"?</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/what-is-rhythm.html">What is Rhythm?</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/staccato.html">Staccato</a></li></ul><p>2. <a href="http://contentspiano.blogspot.com/2006/02/cycling-chopins-fantaisie-impromptu.html">Cycling (Chopin's Fantaisie Impromptu)</a> </p><p>3. Trills & Tremolos </p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/trills.html">Trills</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/tremolos-beethovens-pathetique-1st.html">Tremolos (Beethoven's Pathetique, 1st Movement)</a> </li></ul><p>4. Hand, Body Motions for Technique</p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/hand-motions.html">Hand Motions</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/playing-with-flat-fingers.html">Playing with Flat Fingers</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/body-motions.html">Body Motions</a></li></ul><p>5. Playing Fast: Scales, Arpeggios and Chromatic Scales (Chopin's Fantaisie Impromptu, Beethoven's Moonlight, 3rd Movement)</p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/scales-thumb-under-thumb-over.html">Scales: Thumb Under, Thumb Over </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/to-motion-explanation-and-video.html">The TO Motion, Explanation and Video </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/practicing-to-speed.html">Practicing TO, Speed </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/scales-origin-nomenclature-and.html">Scales: Origin, Nomenclature, and Fingerings </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/arpeggios-chopin-cartwheel-motion.html">Arpeggios (Chopin, Cartwheel Motion)</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/thrust-and-pull-beethovens-moonlight.html">Thrust and Pull, Beethoven's Moonlight, 3rd Movement </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/thumb-most-versatile-finger-examples.html">Thumb: the Most Versatile Finger; Examples of Scale/Arpeggio Practice Routines </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/fast-chromatic-scales.html">Fast Chromatic Scales </a></li></ul><p>6. Memorizing</p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/why-memorize.html">Why Memorize?</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/who-can-what-to-and-when-to-memorize.html">Who can, What to, and When to, Memorize </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/memorizing-and-maintenance.html">Memorizing and Maintenance </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/hand-memory.html">Hand Memory </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/starting-memorizing-process.html">Starting the Memorizing Process </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/reinforcing-memory.html">Reinforcing the Memory </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/practicing-cold.html">Practicing Cold </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/slow-play.html">Slow Play </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/mental-timing.html">Mental Timing </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/establishing-permanent-memory-and.html">Establishing Permanent Memory </a>
1. Hand Memory
2. Music Memory
3. Photographic Memory
4. Keyboard Memory -- Mental Play
5. Theoretical Memory </li>
<li><a href="http://contentspiano.blogspot.com/2006/02/maintenance.html">Maintenance</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/sight-readers-versus-memorizers.html">Sight Readers versus Memorizers:</a> Learning Bach's Inventions; Inventions #1, #8, #13; Quiet Hands; Sinfonia #15 </li><li><a href="http://contentspiano.blogspot.com/2006/02/human-memory-function.html">Human Memory Function </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/how-to-become-good-memorizer.html">How to Become a Good Memorizer </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/summary_09.html">Summary</a></li></ul><p>7. Exercises</p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/introduction-intrinsic-limbering-and.html">Introduction: Intrinsic, Limbering, and Conditioning Exercises </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/parallel-set-exercises-for-intrinsic.html">Parallel Set Exercises for Intrinsic Technical Development </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/how-to-use-parallel-set-exercises.html">How To Use The Parallel Set Exercises </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/scales-arpeggios-finger-independence.html">Scales, Arpeggios, Finger Independence and Finger Lifting Exercises </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/playing-wide-chords-palm-stretching.html">Playing (Wide) Chords, Palm Stretching Exercises </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/practicing-jumps.html">Practicing Jumps </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/stretching-and-other-exercises.html">Stretching and Other Exercises </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/problems-with-hanon-exercises.html">Problems with Hanon Exercises </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/practicing-for-speed.html">Practicing for Speed</a></li></ul><p>8. <a href="http://contentspiano.blogspot.com/2006/02/outlining-beethovens-sonata-1.html">Outlining (Beethoven's Sonata #1)</a> </p><p>9. <a href="http://contentspiano.blogspot.com/2006/02/polishing-piece-eliminating-flubs.html">Polishing a Piece - Eliminating Flubs </a></p><p>10. <a href="http://contentspiano.blogspot.com/2006/02/cold-hands-illness-hand-injury-ear.html">Cold Hands, Illness, Injury, Ear Damage </a></p><p>11. <a href="http://contentspiano.blogspot.com/2006/02/sight-reading.html">Sight Reading </a></p><p>12. <a href="http://contentspiano.blogspot.com/2006/02/learning-relative-pitch-and-perfect.html">Learning Relative Pitch and Perfect Pitch (Sight Singing)</a> </p><p>13. <a href="http://contentspiano.blogspot.com/2006/02/videotaping-and-recording-your-own.html">Videotaping and Recording Your Own Playing </a></p><p>14. Preparing for Performances and Recitals </p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/benefits-and-pitfalls-of.html">Benefits and Pitfalls of Performances/Recitals </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/basics-of-flawless-performances.html">Basics of Flawless Performances </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/practicing-for-performances.html">Practicing for Performances </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/practicing-musically.html">Practicing Musically </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/casual-performances.html">Casual Performances</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/performance-preparation-routines.html">Performance Preparation Routines </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/during-recital.html">During the Recital </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/that-unfamiliar-piano.html">That Unfamiliar Piano </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/after-recital.html">After the Recital</a></li></ul><p>15. <a href="http://contentspiano.blogspot.com/2006/02/origin-and-control-of-nervousness.html">Origin and Control of Nervousness </a></p><p>16. Teaching </p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/types-of-teachers.html">Types of Teachers </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/teaching-youngsters-parental.html">Teaching Youngsters, Parental Involvement </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/memorizing-reading-theory-mental-play.html">Reading, Memorizing, Theory, Mental Play, Absolute Pitch </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/some-elements-of-piano-lessons.html">Some Elements of Piano Lessons </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/why-greatest-pianists-could-not-teach.html">Why the Greatest Pianists Could Not Teach </a></li></ul><p>17. Uprights, Grands, & Electronics, Purchasing and Care </p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/grand-upright-or-electronic.html">Grands, Uprights, or Electronics?</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/electronic-pianos.html">Electronic Pianos </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/uprights.html">Uprights</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/grands.html">Grands</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/purchasing-acoustic-piano.html">Purchasing an Acoustic Piano </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/piano-care.html">Piano Care </a></li></ul><p>18. How to Start Learning Piano: Youngest Children to Old Adults</p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/do-you-need-teacher.html">Do You Need a Teacher?</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/starter-books-and-keyboards.html">Starter Books and Keyboards </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/beginners-age-0-to-65.html">Beginners: Age 0 to 65+</a></li></ul><p>19. <a href="http://contentspiano.blogspot.com/2006/02/ideal-practice-routine-bachs-teachings.html">The “Ideal” Practice Routine (Bach’s Teachings and Invention #4)</a> </p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/learning-rules.html">Learning the Rules</a> </li><li><a href="http://contentspiano.blogspot.com/2006/02/routine-for-learning-new-piece.html">Routine for Learning a New Piece (Invention #4) </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/normal-practice-routines-and-bachs.html">"Normal” Practice Routines and Bach’s Teachings </a></li></ul><p>20. <a href="http://contentspiano.blogspot.com/2006/02/bach-greatest-composer-and-teacher-15.html">Bach: the Greatest Composer and Teacher (15 Inventions and their parallel sets) </a></p><p>21. <a href="http://contentspiano.blogspot.com/2006/02/psychology-of-piano.html">The Psychology of Piano </a></p><p>22. <a href="http://contentspiano.blogspot.com/2006/02/summary-of-method.html">Summary of Method </a></p><p><strong><span style="font-size:130%;">Mathematical Theory of Piano Playing</span></strong></p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/why-do-we-need-mathematical-theory.html">Why do we need a Mathematical Theory?</a> </li><li>The Theory of Finger Motion for Piano
a. <a href="http://contentspiano.blogspot.com/2006/02/serial-parallel-play.html">Serial, Parallel Play </a>
b. <a href="http://contentspiano.blogspot.com/2006/02/speed-walls.html">Speed Walls </a>
c. <a href="http://contentspiano.blogspot.com/2006/02/increasing-speed.html">Increasing Speed</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/thermodynamics-of-piano-playing.html">Thermodynamics of Piano Playing </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/mozarts-formula-beethoven-and-group.html">Mozart's Formula, Beethoven and Group Theory</a>
a. <a href="http://contentspiano.blogspot.com/2006/02/mozart-eine-kleine-nachtmusik-sonata.html">Mozart: Eine Kleine Nachtmusik, Sonata K300 </a>
b. <a href="http://contentspiano.blogspot.com/2006/02/beethoven-5th-symphony-appassionata.html">Beethoven: 5th Symphony, Appassionata, Waldstein</a></li><li><a href="http://contentspiano.blogspot.com/2006/02/learning-rate-calculation.html">Learning Rate Calculation </a>(1000 Times Faster!) </li><li><a href="http://contentspiano.blogspot.com/2006/02/future-research-topics.html">Future Research Topics </a>
a. Momentum Theory of Piano Playing
b. The Physiology of Technique
c. Brain Research (HS vs HT Play, etc.)
d. What Causes Nervousness?
e. Causes of and Remedies for Tinnitus
f. What is Music?
g. At What Age to Start Piano?
h. The Future of Piano
i. The Future of Education </li></ul><p></p><p></p><p align="center"><strong>Chapter 2</strong></p><p align="center"><strong><span style="font-size:180%;">Tuning Your Piano</span></strong><span style="font-size:130%;">
</span><span style="font-size:100%;"></span></p><p align="left"><span style="font-size:100%;">1. <a href="http://contentspiano.blogspot.com/2006/02/introduction.html">Introduction</a> </span><span style="font-size:100%;">
2. <a href="http://contentspiano.blogspot.com/2006/02/chromatic-scale-and-temperament.html">Chromatic Scale and Temperament </a></span></p><ul><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/mathematics-of-chromatic-scale-and.html">Mathematics of the Chromatic Scale and Chords </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/temperament-music-and-circle-of-fifths.html">Temperaments, Music, and the Circle of Fifths </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/pythagorean-equal-meantone-and-well.html">Pythagorean, Equal, Meantone, and “Well” Temperaments </a></span></li></ul><p><span style="font-size:100%;">3. <a href="http://contentspiano.blogspot.com/2006/02/tuning-tools.html">Tuning Tools</a>
4. <a href="http://contentspiano.blogspot.com/2006/02/preparation.html">Preparation</a>
5. <a href="http://contentspiano.blogspot.com/2006/02/getting-started.html">Getting Started </a></span></p><ul><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/engaging-and-manipulating-tuning-lever.html">Engaging and Manipulating the Tuning Lever </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/setting-pin.html">Setting the Pin </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/tuning-unisons.html">Tuning Unisons </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/sympathetic-vibrations.html">Sympathetic Vibrations </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/making-that-final-infinitesimal-motion.html">Making that Final Infinitesimal Motion </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/equalizing-string-tension.html">Equalizing String Tensions </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/rocking-it-in-treble.html">Rocking It in the Treble </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/rumblings-in-bass.html">Rumblings in the Bass </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/harmonic-tuning.html">Harmonic Tuning </a></span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/what-is-stretch.html">What is Stretch?</a> </span></li><li><span style="font-size:100%;"><a href="http://contentspiano.blogspot.com/2006/02/precision-precision-precision.html">Precision, Precision, Precision </a></span></li></ul><span style="font-size:100%;"><p>6. <a href="http://contentspiano.blogspot.com/2006/02/tuning-procedures-and-temperament.html">Tuning Procedures and Temperament </a></p><ul><li><a href="http://contentspiano.blogspot.com/2006/02/tuning-piano-to-tuning-fork.html">Tuning the Piano to the Tuning Fork </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/kirnberger-ii.html">Kirnberger II </a></li><li><a href="http://contentspiano.blogspot.com/2006/02/equal-temperament.html">Equal Temperament</a></li></ul><p>7. <a href="http://contentspiano.blogspot.com/2006/02/making-minor-repairs-voicing-and.html">Making Minor Repairs (Voicing and Polishing the Capstans)</a> </p><ul><li>Hammer Voicing </li><li>Polishing the Capstans </li></ul><p align="center"><strong>Chapter 3</strong></p><p align="center"></span><strong><span style="font-size:180%;">Scientific Method, Theory of Learning and the Brain</span></strong></p><p align="left">1. <a href="http://fundamentalpiano3.blogspot.com/2006/03/introduction.html">Introduction </a>
2. <a href="http://fundamentalpiano3.blogspot.com/2006/03/scientific-approach.html">The Scientific Approach </a>
3. <a href="http://fundamentalpiano3.blogspot.com/2006/03/what-is-scientific-method.html">What is a scientific Method?</a>
4. <a href="http://fundamentalpiano3.blogspot.com/2006/03/theory-of-learning.html">Theory of Learning </a>
5. <a href="http://fundamentalpiano3.blogspot.com/2006/03/what-causes-dreams-and-methods-for.html">What Causes Dreams and Methods for Controlling Them </a></p><ul><li><a href="http://fundamentalpiano3.blogspot.com/2006/03/falling-dream.html">The Falling Dream </a></li><li><a href="http://fundamentalpiano3.blogspot.com/2006/03/inability-to-run-dream.html">Inability-to-Run Dream </a></li><li><a href="http://fundamentalpiano3.blogspot.com/2006/03/late-to-exams-or-getting-lost-dream.html">Late-to-Exam or Getting-Lost Dream </a></li><li><a href="http://fundamentalpiano3.blogspot.com/2006/03/solving-my-long-and-complex-dream.html">Solving my Long and Complex Dream </a></li><li><a href="http://fundamentalpiano3.blogspot.com/2006/03/controlling-dreams.html">Controlling Dreams </a></li><li><a href="http://fundamentalpiano3.blogspot.com/2006/03/what-these-dreams-teach-us-about-our.html">What Dreams Tell Us about our Brains </a></li></ul><p>6. <a href="http://fundamentalpiano3.blogspot.com/2006/03/how-to-use-your-subconscious-brain.html">How to Use Your Subconscious Brain</a></p><ul><li><a href="http://fundamentalpiano3.blogspot.com/2006/03/emotions.html">Emotions</a></li><li><a href="http://fundamentalpiano3.blogspot.com/2006/03/using-subconscious-brain.html">Using the Subconscious Brain </a></li></ul><p></p><p></p><p><a href="http://contentspiano.blogspot.com/2006/02/testimonials.html"><strong>TESTIMONIALS</strong></a></p><p><a href="http://contentspiano.blogspot.com/2006/02/preface.html"><strong>PREFACE</strong></a></p><p><a href="http://contentspiano.blogspot.com/2005/10/abbreviations-frequently-used-phrases.html">ABBREVIATIONS &amp; Frequently Used Phrases </a></p>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com2tag:blogger.com,1999:blog-22149718.post-1140803548994587492006-02-24T09:52:00.000-08:002006-02-24T09:54:27.880-08:00Making Minor Repairs (Voicing and Polishing the Capstans)<P>Once you start tuning, you cannot help but get involved in small repairs and conducting some maintenance.</P>
<B><P>a. Hammer Voicing</P></B>
<B><P>A common problem seen with many pianos is compacted hammers. I raise this point because the condition of the hammer is much more important to the proper development of piano technique and for cultivating performance skills, than many people realize.</B></I> Numerous places in this book refer to the importance of practicing musically in order to acquire technique. But you can't play musically if the hammer can't do its job, a critical point that is overlooked even by many tuners (often because they are afraid that the extra cost will drive customers away). For a grand piano, a sure sign of compacted hammers is that you find the need to close the lid at least partially in order to play soft passages. Another sure sign is that you tend to use the soft pedal to help you play softly. Compacted hammers either give you a loud sound or none at all. Each note tends to start with an annoying percussive bang that is too strong, and the sound is overly bright. It is these percussive bangs that are so damaging to the tuners' ear. A properly voiced piano enables control over the entire dynamic range and produces a more pleasing sound.</P>
<P>Let's first see how a compacted hammer can produce such extreme results. How do small, light hammers produce loud sounds by striking with relatively low force on strings under such high tension? If you were to try to push down on the string or try to pluck it, you will need quite a large force just to make a small sound. The answer lies in an incredible phenomenon that occurs when tightly stretched strings are struck at right angles to the string. <B><I>It turns out that the force produced by the hammer at the instant of impact is theoretically infinite!</B></I> This nearly infinite force is what enables the light hammer to overcome practically any achievable tension on the string and cause it to vibrate.</P>
<P>Here is the calculation for that force. Imagine that the hammer is at its highest point after striking the string (grand piano). The string at this point in time makes a triangle with its original horizontal position (this is just an idealized approximation, see below). The shortest leg of this triangle is the length between the agraffe and the impact point of the hammer. The second shortest leg is from the hammer to the bridge. The longest is the original horizontal configuration of the string, a straight line from bridge to agraffe. Now if we drop a vertical line from the hammer strike point down to the original string position, we get two right triangles back-to-back. These are two extremely skinny right triangles that have very small angles at the agraffe and at the bridge; we will call these small angles "theta"s.</P>
<P>The only thing we know at this time is the force of the hammer, but this is not the force that moves the string, because the hammer must overcome the string tension before the string will yield. That is, the string cannot move up unless it can elongate. This can be understood by considering the two right triangles described above. The string had the length of the long legs of the right triangles before the hammer struck, but after the strike, the string is the hypotenuse, which is longer. That is, if the string were absolutely inelastic and the ends of the string were rigidly fixed, no amount of hammer force will cause the string to move.</P>
<P>It is a simple matter to show, using vector diagrams, that the <I>extra</I> tension force F (in addition to the original string tension) produced by the hammer strike is given by f = Fsin(theta), where f is the force of the hammer. It does not matter which right triangle we use for this calculation (the one on the bridge side or on the agraffe side). Therefore, the string tension F = f/sin(theta). At the initial moment of the strike, theta = 0, and therefore F = infinity! This happens because sin(0) = 0. Of course, F can get to infinity only if the string cannot stretch and nothing else can move. What happens in reality is that as F increases towards infinity, something gives (the string stretches, the bridge moves, etc.) so that the hammer begins to move the string and theta increases from zero, making F finite.</P>
<P>This force multiplication explains why a small child can produce quite a loud sound on the piano in spite of the hundreds of pounds of tension on the strings. It also explains why an ordinary person can break a string just playing the piano, especially if the string is old and has lost its elasticity. The lack of elasticity causes the F to increase far more than if the string were more elastic, the string cannot stretch, and theta remains close to zero. This situation is greatly exacerbated if the hammer is also compacted so that there is a large, flat, hard groove that contacts the string. In that case, the hammer surface has no give and the instantaneous "f" in the above equation becomes very large. Since all this happens near theta = 0 for a compacted hammer, the force multiplication factor is also increased. The result is a broken string.</P>
<P>The above calculation is a gross over-simplification and is correct only qualitatively. In reality, a hammer strike initially throws out a traveling wave towards the bridge, similarly to what happens when you grab one end of a rope and flick it. The way to calculate such waveforms is to solve certain differential equations that are well known. The computer has made the solution of such differential equations a simple matter and realistic calculations of these waveforms can now be made routinely. Therefore, although the above results are not accurate, they give a qualitative understanding of what is happening, and what the important mechanisms and controlling factors are.</P>
<P>For example, the above calculation shows that it is not the transverse vibration energy of the string, but the tensile force on the string, that is responsible for the piano sound. The energy imparted by the hammer is stored in the entire piano, not just the strings. This is quite analogous to the bow and arrow -- when the string is pulled, all the energy is stored in the bow, not the string. And all of this energy is transferred via the tension in the string. In this example, the mechanical advantage and force multiplication calculated above (near theta = 0) is easy to see. It is the same principle on which the harp is based.</P>
<P>The easiest way to understand why compacted hammers produce higher harmonics is to realize that the impact occurs in a shorter time. When things happen faster, the string generates higher frequency components in response to the faster event.</P>
<P>The above paragraphs make it clear that a compacted hammer will produce a large initial impact on the string whereas a properly voiced hammer will be much gentler on the string thus imparting more of its energy to the lower frequencies than the harmonics. Because the same amount of energy is dissipated in a shorter amount of time for the compacted hammer, the instantaneous sound level can be much higher than for a properly voiced hammer, especially at the higher frequencies. Such short sound spikes can damage the ear without causing any pain. Common symptoms of such damage are tinnitus (ringing in the ear) and hearing loss at high frequencies. Piano tuners, when they must tune a piano with such worn hammers, would be wise to wear ear plugs. It is clear that voicing the hammer is at least as important as tuning the piano, especially because we are talking about potential ear damage. An out-of-tune piano with good hammers does not damage the ear. Yet many piano owners will have their pianos tuned but neglect the voicing. </P>
<B><I><P>The two most important procedures in voicing are hammer re-shaping and needling. </P>
</B></I><P>When the flattened strike point on the hammer exceeds about 1 cm, it is time to re-shape the hammer. Note that you have to distinguish between the string groove length and flattened area; even in hammers with good voicing, the grooves may be over 5 mm long. In the final analysis you will have to judge on the basis of the sound. Shaping is accomplished by shaving the "shoulders" of the hammer so that it regains its previous rounded shape at the strike point. It is usually performed using 1 inch wide strips of sandpaper attached to strips of wood or metal with glue or double sided tape. You might start with 80 grit garnet paper and finish it off with 150 grit garnet paper. The sanding motion must be in the plane of the hammer; never sand across the plane. There is almost never a need to sand off the strike point. Therefore, leave about 2 mm of the center of the strike point untouched.</P>
<P>Needling is not easy because the proper needling location and needling depth depend on the particular hammer (manufacturer) and how it was originally voiced. Especially in the treble, hammers are often voiced at the factory using hardeners such as lacquer, etc. Needling mistakes are generally irreversible. Deep needling is usually required on the shoulders just off the strike point. Very careful and shallow needling of the strike point area may be needed. The tone of the piano is extremely sensitive to shallow needling at the strike point, so that you must know exactly what you are doing. When properly needled, the hammer should allow you to control very soft sounds as well as produce loud sounds without harshness. You get the feeling of complete tonal control. You can now open your grand piano fully and play very softly without the soft pedal! You can also produce those loud, rich, authoritative tones.</P>
<B><P>b. Polishing the Capstans</P>
</B><P>Polishing the capstans can be a rewarding maintenance procedure. They may need polishing if they have not been cleaned in over 10 years, sometimes sooner. Press down on the keys slowly to see if you can feel a friction in the action. A frictionless action will feel like sliding an oily finger along a smooth glassware. When friction is present, it feels like the motion of a clean finger on squeaky clean glass. In order to be able to get to the capstans, you will need to lift the action off from the keys by unscrewing the screws that hold the action down for the grand. For uprights you generally need to unscrew the knobs that hold the action in place; make sure that the pedal rods, etc., are disengaged.</P>
<P>When the action is removed, the keys can be lifted out after removing the key stop rail. First make sure that all the keys are numbered so that you can replace them in the correct order. This is a good time to remove all the keys and clean any previously inaccessible areas as well as the sides of the keys. You can use a mild cleaning agent such as a cloth dampened with Windex for cleaning the sides of the keys.</P>
<P>See if the top, spherical contact areas of the capstans are tarnished. If they do not have a shiny polish, they are tarnished. Use any good brass/bronze/copper polish (such as Noxon) to polish and buff up the contact areas. Reassemble, and the action should now be much smoother.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803503179826522006-02-24T09:51:00.000-08:002006-02-24T09:51:43.180-08:00Equal Temperament<P>I present here the easiest, approximate, equal temperament scheme. More accurate algorithms can be found in the literature (Reblitz, Jorgensen). No self-respecting professional tuner would use this scheme; however, when you get good at it, you can produce a decent equal temperament. For the beginner, the more complete and precise schemes will not necessarily give better results. With the more complex methods, a beginner can quickly get confused without any idea of what he did wrong. With the method shown here, you can quickly develop the ability to find out what you did wrong.</P>
<P>Mute the side strings from G3 to C#5. Tune A4 to the A440 fork. Tune A3 to A4. Then tune up in contracted 5ths from A3 until you cannot go up any more without leaving the muted range, then tune one octave down, and repeat this up-in-5ths and down-one-octave procedure until you get to A4. For example, you will start with a contracted A3-E4, then a contracted E4-B4. The next 5th will take you above the highest muted note, C#4, so you tune one octave down, B4-B3. All octaves are, of course, just. The contracted 5ths should beat a little under 1 Hz at the bottom of the muted range and about 1.5 Hz near the top. The beat frequencies of the 5ths between these highest and lowest limits should increase smoothly with increasing pitch. </P>
<P>When going up in 5ths, you tune flat from just to create a contracted 5th. Therefore you can start from just and tune flat in order to increase the beat frequency to the desired value and set the pin correctly at the same time. If you had done everything perfectly, the last D4-A4 should be a contracted 5th with a beat frequency of 1 Hz without any tuning. Then, you are done. You have just done a "circle of 5ths". The miracle of the circle of 5ths is that it tunes every note once, without skipping any within the A3-A4 octave!</P>
<P>If the final D4-A4 is not correct, you made some errors somewhere. In that case, reverse the procedure, starting from A4, going down in contracted 5ths and up in octaves, until you reach A3, where the final A3-E4 should be a contracted 5th with a beat frequency slightly under 1 Hz. For going down in 5ths, you create a contracted 5th by tuning sharp from just. However, this tuning action will not set the pin. Therefore, in order to set the pin correctly, you must first go too sharp, and then decrease the beat frequency to the desired value. Therefore, going down in 5ths is a more difficult operation than going up in 5ths.</P>
<P>An alternative method is to start with A and tune to C by going up in 5ths, and checking this C with a tuning fork. If your C is too sharp, your 5ths were not sufficiently contracted, and vice versa. Another variation is to tune up in 5ths from A3 a little over half way, and then tune down from A4 to the last note that you tuned coming up. </P>
<P>Once the bearings are set, continue as described in the Kirnberger section above.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803294463491502006-02-24T09:48:00.001-08:002006-03-16T21:58:12.163-08:00What is Stretch?<p>Harmonic tuning is always associated with a phenomenon called stretch. Harmonics in piano strings are never exact because real strings attached to real ends do not behave like ideal mathematical strings. This property of inexact harmonics is called inharmonicity. The difference between the actual and theoretical harmonic frequencies is called stretch. Experimentally, it is found that most harmonics are sharp compared to their ideal theoretical values, although there can be a few that are flat. </p><p>According to one research result (Young, 1952), stretch is caused by inharmonicity due to the stiffness of strings. Ideal mathematical strings have zero stiffness. Stiffness is what is called an extrinsic property -- it depends on the dimensions of the wire. If this explanation is correct, then stretch must also be extrinsic. Given the same type of steel, the wire is stiffer if it is fatter or shorter. One consequence of this dependence on stiffness is an increase in the frequency with harmonic mode number; i.e., the wire appears stiffer to harmonics with shorter wavelengths. Stiffer wires vibrate faster because they have an extra restoring force, in addition to the string tension. This inharmonicity has been calculated to within several percent accuracy so that the theory appears to be sound, and this single mechanism appears to account for most of the observed stretch. </p><p>These calculations show that stretch is about 1.2 cents for the second mode of vibration at C4 and doubles about every 8 semitones at higher frequency (C4 = middle C, the first mode is the lowest, or fundamental frequency, one cent is one hundredth of a semitone, and there are 12 semitones in an octave). The stretch becomes smaller for lower notes, especially below C3, because the wire wound strings are quite flexible. Stretch increases rapidly with mode number and decreases even more rapidly with string length. In principle, stretch is smaller for larger pianos and larger for lower tension pianos if the same diameter strings are used. Stretch presents problems in scale design since abrupt changes in string type, diameter, length, etc., will produce a discontinuous change in stretch. Very high mode harmonics, if they happen to be unusually loud, present problems in tuning because of their large stretch -- tuning out their beats could throw the lower, more important, harmonics audibly out of tune. </p><p>Since larger pianos tend to have smaller stretch, but also tend to sound better, one might conclude that smaller stretch is better. However, the difference in stretch is generally small, and the tone quality of a piano is largely controlled by properties other than stretch. </p><p>In harmonic tuning you tune, for example, the fundamental or a harmonic of the upper note to a higher harmonic of the lower note. The resulting new note is not an exact multiple of the lower note, but is sharp by the amount of stretch. What is so interesting about stretch is that a scale with stretch produces "livelier" music than one without! This has caused some tuners to tune in double octaves instead of single octaves, which increases the stretch. </p><p>The amount of stretch is unique to each piano and, in fact, is unique to each note of each piano. Modern electronic tuning aids are sufficiently powerful to record the stretch for all the desired notes of individual pianos. Tuners with electronic tuning aids can also calculate an average stretch for each piano or stretch function and tune the piano accordingly. In fact, there are anecdotal accounts of pianists requesting stretch in excess of the natural stretch of the piano. In aural tuning, stretch is naturally, and accurately, taken into account. Therefore, although stretch is an important aspect of tuning, the tuner does not have to do anything special to include stretch, if all you want is the natural stretch of the piano.</p>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803323090224842006-02-24T09:48:00.000-08:002006-02-24T09:48:43.093-08:00Precision, Precision, Precision<I><B><P>The name of the game in tuning is precision.</B></I> All tuning procedures are arranged in such a way that you tune the first note to the tuning fork, the second to the first, etc., in sequence. Therefore, any errors will quickly add up. In fact, an error at one point will often make some succeeding steps impossible. This happens because you are listening for the smallest hint of beats and if the beats were not totally eliminated in one note, you can't use it to tune another as those beats will be clearly heard. In fact, for beginners, this will happen frequently before you learn how precise you need to be. When this happens, you will hear beats that you can't eliminate. In that case, go back to your reference note and see if you hear the same beat; if you do, there is the source of your problem -- fix it.</P>
<B><I><P>The best way to assure precision is by checking the tuning.</B></I> Errors occur because every string is different and you are never sure that the beat you hear is the one you are looking for, especially for the beginner. Another factor is that you need to count beats per second (bps), and your idea of, say 2bps, will be different on different days or at different times of the same day until you have those "beat speeds" well memorized. Because of the critical importance of precision, it pays to check each tuned note. This is especially true when "setting the bearings" which is explained below. Unfortunately, it is just as difficult to check as it is to tune correctly; that is, a person who cannot tune sufficiently accurately is usually unable to perform a meaningful check. In addition, if the tuning is sufficiently off, the checking doesn't work. Therefore, <B><I>I have provided methods of tuning below that use a minimum of checks.</B></I> The resulting tuning will not be very good initially, for equal temperament. The Kirnberger temperament (see below) is easier to tune accurately. On the other hand, beginners can't produce good tunings anyway, no matter what methods they use. At least, the procedures presented below will provide a tuning which should not be a disaster and which will improve as your skills improve. <B><I>In fact, the procedure described here is probably the fastest way to learn.</B></I> After you have improved sufficiently, you can then investigate the checking procedures, such as those given in Reblitz, or "Tuning" by Jorgensen. </P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803246719106002006-02-24T09:47:00.000-08:002006-02-24T09:47:26.720-08:00Harmonic Tuning<P>Once you are satisfied with your ability to tune unisons, start practicing tuning octaves. Take any octave near middle C and mute out the upper two side strings of each note by inserting a wedge between them. Tune the upper note to the one an octave below, and vice versa. As with unisons, start near middle C, then work up to the highest treble, and then practice in the bass. Repeat the same practice with 5ths, 4ths, and major 3rds. </P>
<B><I><P>After you can tune perfect harmonics, try de-tuning to see if you can hear the increasing beat frequency as you deviate very slightly from perfect tune.</B></I> Try to identify various beat frequencies, especially 1bps (beat per second) and 10bps, using 5ths. These skills will come in handy later.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803215166889262006-02-24T09:46:00.001-08:002006-02-24T09:46:55.166-08:00Rumblings in the Bass<I><B><P>The lowest bass strings are second in difficulty (to the highest notes) to tune.</B></I> These strings produce sound composed mostly of higher harmonics. Near the tuning point, the beats are so slow and soft that they are difficult to hear. Sometimes, you can "hear" them better by pressing your knee against the piano to feel for the vibrations than by trying to hear them with your ears, especially in the single string section. You can practice unison tuning only down to the last double string section. <B><I>See if you can recognize the high pitched, metallic, ringing beats that are prevalent in this region.</B></I> Try eliminating these and see if you need to de-tune slightly in order to eliminate them. If you can hear these high, ringing, beats, it means that you are well on your way. Don't worry if you can't even recognize them at first-- beginners are not expected to.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803187751457002006-02-24T09:46:00.000-08:002006-02-24T09:46:27.753-08:00Rocking It in the Treble<I><B><P>The most difficult notes to tune are the highest ones.</B></I> Here you need incredible accuracy in moving the strings and the beats are difficult to hear. Beginners can easily lose their bearing and have a hard time finding their way back. One advantage of the need for such small motions is that now, you can use the pin-rocking motion to tune. Since the motion is so small, rocking the pin does not damage the pinblock. <B><I>To rock the pin, place the lever parallel to the strings and pointing towards the strings (away from you). To tune sharp, pull up on the lever, and to tune flat, press down.</B></I> First, make sure that the tuning point is close to the center of the rocking motion. If it is not, rotate the pin so that it is. Since this rotation is much larger than that needed for the final tuning, it is not difficult, but remember to correctly set the pin. It is better if the tuning point is front of center (towards the string), but bringing it too far forward would risk damaging the pinblock when you try to tune flat. Note that tuning sharp is not as damaging to the pinblock as tuning flat because the pin is already jammed up against the front of the hole.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803136184149222006-02-24T09:45:00.000-08:002006-02-24T09:45:51.513-08:00Equalizing String Tension<I><B><P>Pounding is also helpful for distributing the string tension more evenly among all the non-speaking sections of the string, such as the duplex scale section, but especially in the section between the capo bar and the agraffe.</B></I> There is controversy as to whether equalizing the tension will improve the sound. There is little question that the even tension will make the tuning more stable. However, whether it makes a <I>material</I> difference in stability may be debatable, especially if the pins were correctly set during tuning. In many pianos, the duplex sections are almost completely muted out using felts because they might cause undesirable oscillations. In fact, the over-strung section is muted out in almost every piano. Beginners need not worry about the tension in these "non-speaking" sections of the strings. Thus heavy pounding, though a useful skill to learn, is not necessary for a beginner.</P>
<P><B><I>My personal opinion is that the sound from the duplex scale strings does not add to the piano sound.</I></B> In fact, this sound is inaudible and is muted out when they become audible in the bass. Thus the “art of tuning the duplex scale” is a myth although most piano tuners (including Reblitz!) have been taught to believe it by the manufacturers, because it makes for a good sales pitch. The only reason why you want to tune the duplex scale is that the bridge wants to be at a node of both the speaking and non-speaking lengths; otherwise, tuning becomes difficult, sustain may be shortened, and you lose uniformity. Using mechanical engineering terminology, we can say that tuning the duplex scale optimizes the vibrational impedance of the bridge. In other words, the myth does not detract from the tuners’ ability to do their job. Nonetheless, a proper understanding is certainly preferable. The duplex scale is needed to allow the bridge to move more freely, not for producing sound. Obviously, the duplex scale will improve the quality of the sound (from the speaking lengths) because it optimizes the impedance of the bridge, but not because it produces any sound. The facts that the duplex scale is muted out in the bass and is totally inaudible in the treble prove that the sound from the duplex scale is not needed. Even in the inaudible treble, the duplex scale is “tuned” in the sense that the aliquot bar is placed at a location such that the length of the duplex part of the string is a harmonic length of the speaking section of the string in order to optimize the impedance (“aliquot” means fractional or harmonic). If the sound from the duplex scale were audible, the duplex scale would have to be tuned as carefully as the speaking length. However, for impedance matching, the tuning need only be approximate, which is what is done in practice. Some manufacturers have stretched this duplex scale myth to ridiculous lengths by claiming a second duplex scale on the pin side. Since the hammer can only transmit tensile strain to this length of string (because of the rigid Capo bar), this part of the string cannot vibrate to produce sound. Consequently, practically no manufacturer specifies that the non-speaking lengths on the pin side be tuned.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803099240285002006-02-24T09:44:00.002-08:002006-02-24T09:44:59.243-08:00Making that Final Infinitesimal Motion<P>We now advance to the next level of difficulty. Find a note near G5 that is slightly out of tune, and repeat the above procedure for G3. The tuning motions are now much smaller for these higher notes, making them more difficult. In fact you may not be able to achieve sufficient accuracy by rotating the pin. We need to learn a new skill. <B><I>This skill requires you to pound on the notes, so put on your ear muffs or ear plugs.</P>
</B></I><P>Typically, you would get through motion (4) successfully, but for motion (5) the pin would either not move or jump past the tuning point. <B><I>In order to make the string advance in smaller increments, press on the lever at a pressure slightly below the point at which the pin will jump. Now strike hard on the note while maintaining the same pressure on the lever. </B></I>The added string tension from the hard hammer blow will advance the string by a small amount. Repeat this until it is in perfect tune. It is important to never release the pressure on the lever and to keep the pressure constant during these repeated small advances, or you will quickly lose track of where you are. When it is in perfect tune, and you release the lever, the pin might spring back, leaving the string slightly flat. You will have to learn from experience, how much it will spring back and compensate for it during the tuning process.</P>
<P>The need to pound on the string to advance it is one reason you often hear tuners pounding on the piano. It is a good idea to get into the habit of pounding on most of the notes because this stabilizes the tuning. The resulting sound can be so loud as to damage the ear, and one of the occupational hazards of tuners is ear damage from pounding. Use of ear plugs is the solution. When pounding, you will still easily hear the beats even with ear plugs. The most common initial symptom of ear damage is tinnitus (ringing in the ear). You can minimize the pounding force by increasing the pressure on the lever. Also, less pounding is required if the lever is parallel to the string instead of perpendicular to it, and even less if you point it to the left. This is another reason why many tuners use their levers more parallel to the strings than perpendicular. Note that there are two ways to point it parallel: towards the strings (12 o'clock) and away from the strings (6 o'clock). As you gain experience, experiment with different lever positions as this will give you many options for solving various problems. For example, with the most popular 5-degree head on your lever, you may not be able to point the lever handle to the right for the highest octave because it may hit the wooden piano frame.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803077086056552006-02-24T09:44:00.001-08:002006-02-24T09:44:37.086-08:00Sympathetic Vibrations<P>The accuracy required to bring two strings into perfect tune is so high that it is a nearly impossible job. It turns out that, in practice, this is made easier because <B><I>when the frequencies approach within a certain interval called the "sympathetic vibration range", the two strings change their frequencies towards each other so that they vibrate with the same frequency.</B></I> This happens because the two strings are not independent, but are coupled to each other at the bridge. When coupled, the string vibrating at the higher frequency will drive the slower string to vibrate at a slightly higher frequency, and vice versa. The net effect is to drive both frequencies towards the average frequency of the two. Thus when you tune 1 and 2 unison, you have no idea whether they are in perfect tune or merely within the sympathetic vibration range (unless you are an experienced tuner). In the beginning, you will most likely not be in perfect tune. </P>
<P>Now if you were to try to tune a third string to the two strings in sympathetic vibration, the third string will bring the string closest to it in frequency into sympathetic vibration. But the other string may be too far off in frequency. It will break off the sympathetic vibration, and will sound dissonant. The result is that no matter where you are, you will always hear beats -- the tuning point disappears! It might appear that if the third string were tuned to the average frequency of the two strings in sympathetic vibration, all three should go into sympathetic vibration. This does not appear to be the case unless all three frequencies are in perfect tune. If the first two strings are sufficiently off, a complex transfer of energy takes place among the three strings. Even when the first two are close, there will be higher harmonics that will prevent all beats from disappearing when a third string is introduced. In addition, there are frequent cases in which you cannot totally eliminate all beats because the two strings are not identical. Therefore, a beginner will become totally lost, if he were to try to tune a third string to a pair of strings. <B><I>Until you become proficient at detecting the sympathetic vibration range, always tune one string to one; never one to two.</B></I> In addition, just because you tuned 1 to 2 and 3 to 2, it does not mean that the three strings will sound "clean" together. Always check; if it is not completely "clean", you will need to find the offending string and try again.</P>
<P>Note the use of the term "clean". With enough practice, you will soon get away from listening to beats, but instead, you will be looking for a pure sound that results somewhere within the sympathetic vibration range. This point will depend on what types of harmonics each string produces. In principle, when tuning unisons, you are trying to match the fundamentals. In practice, a slight error in the fundamentals is inaudible compared to the same error in a high harmonic. Unfortunately, these high harmonics are generally not exact harmonics but vary from string to string. Thus, when the fundamentals are matched, these high harmonics create high frequency beats that make the note "muddy" or "tinny". When the fundamentals are de-tuned ever so slightly so that the harmonics do not beat, the note "cleans up". <B><I>Reality is even more complicated because some strings, especially for the lower quality pianos, will have extraneous resonances of their own, making it impossible to completely eliminate certain beats.</B></I> These beats become very troublesome if you need to use this note to tune another one.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803054121029582006-02-24T09:44:00.000-08:002006-02-24T09:44:14.126-08:00Tuning Unisons<P>Now engage the tuning lever on the pin for string 1. We will tune string 1 to string 2. <B><I>The motion you will practice is: (1) flat, (2) sharp, (3) flat, (4) sharp and (5) flat (tune).</B></I> Except for (1), each motion must be smaller than the previous one. As you improve, you will add or eliminate steps as you see fit. We are assuming that the two strings are almost in tune. As you tune, you must follow two rules: <B><I>(a) never turn the pin unless you are simultaneously listening to the sound, and (b) never release the pressure on the tuning lever handle until that motion is complete.</B></I> </P>
<P>For example, let's start with motion (1) flat: keep playing the note every second or two with the LH, so that there is a continuous sound, while pushing the end of the lever handle away from you with the thumb and 2nd finger. Play the note in such a way as to maintain a continuous sound. Don't lift the key for any length of time, as this will stop the sound. Keep the key down and play with a quick up-and-down motion so that there is no break in the sound. The pinky and the rest of your RH should be braced against the piano. The required motion of the lever is just a few millimeters. First, you will feel an increasing resistance, and then the pin will start to rotate. Before the pin begins to rotate, you should hear a change in the sound. As you turn the pin, listen for string 1 going flat, creating a beat with the center string; the beat frequency increasing as you turn. Stop at a beat frequency of 2 to 3 per second. The tip of the tuning lever should move less than one cm. Remember, never rotate the pin when there is no sound because you will immediately lose track of where you are with respect to how the beats are changing. Always maintain constant pressure on the lever until that motion is completed for the same reason.</P>
<P>What is the rationale behind the above 5 motions? Assuming that the two strings are in reasonable tune, you first tune string 1 flat in step (1) to make sure that in step (2) you will pass the tuning point. This also protects against the possibility that you had placed the lever on the wrong tuning pin; as long as you are turning flat, you will never break a string. </P>
<P>After (1) you are flat for sure, so in step (2) you can listen to the tuning point as you pass through it. Go past it until you hear a beat frequency of about 2 to 3 per second on the sharp side, and stop. Now you know where the tuning point is, and what it sounds like. The reason for going so far past the tuning point is that you want to set the pin, as explained above. </P>
<P>Now go back flat again, step (3), but this time, stop just past the tuning point, as soon as you can hear any incipient beats. The reason why you don't want to go too far past the tuning point is that you don't want to undo the "setting of the pin" in step (2). Again, note exactly what the tuning point sounds like. It should sound perfectly clean and pure. This step assures that you did not set the pin too far. </P>
<P>Now conduct the final tuning by going sharp (step 4), by as little as you can beyond perfect tune, and then bringing it into tune by turning flat (step 5). Note that your final motion must always be flat in order to set the pin. Once you become good, you might be able to do the whole thing in two motions (sharp, flat), or three (flat, sharp, flat). </P>
<P>Ideally, from step (1) to final tune, you should maintain the sound with no stoppage, and you should always be exerting pressure on the handle; never letting go of the lever. Initially, you will probably have to do this motion by motion. When you become proficient, the whole operation will take just a few seconds. But at first, it will take <I>a lot</I> longer. Until you develop your "tuning muscles" you will tire quickly and may have to stop from time to time to recover. Not only the hand/arm muscles, but the mental and ear concentration required to focus on the beats can be quite a strain and can quickly cause fatigue. You will need to develop "tuning stamina" gradually. Most people do better by listening through one ear than through both, so turn your head to see which ear is better. </P>
<B><I><P>The most common mistake beginners make at this stage is to try to listen for beats by pausing the tuning motion.</B></I> Beats are difficult to hear when nothing is changing. If the pin is not being turned, it is difficult to decide which of the many things you are hearing is the beat that you need to concentrate on. <B><I>What tuners do is to keep moving the lever and then listening to <U>the changes in the beats</U>.</B></I> When the beats are changing, it is easier to identify the particular beat that you are using for tuning that string. Therefore, slowing down the tuning motion doesn't make it easier. Thus the beginner is between a rock and a hard place. Turning the pin too quickly will result in all hell breaking loose and losing track of where you are. On the other hand, turning too slowly will make it difficult to identify the beats. Therefore work on determining the range of motion you need to get the beats and the right speed with which you can steadily turn the pin to make the beats come and go. In case you get hopelessly lost, mute strings 2 and 3 by placing a wedge between them, play the note and see if you can find another note on the piano that comes close. If that note is lower than G3, then you need to tune it sharp to bring it back, and vice versa.</P>
<P>Now that you have tuned string 1 to string 2, reposition the wedge so that you mute 1, leaving 2 and 3 free to vibrate. Tune 3 to 2. When you are satisfied, remove the wedge and see if the G is now free of beats. You have tuned one note! If the G was in reasonable tune before you started, you haven't accomplished much, so find a note nearby that is out of tune and see if you can "clean it up". Notice that in this scheme, you are always tuning one single string to another single string. In principle, if you are really good, strings 1 and 2 are in perfect tune after you finish tuning 1, so you don't need the wedge any more. You should be able to tune 3 to 1 and 2 vibrating together. In practice this doesn't work until you become really proficient. This is because of a phenomenon called sympathetic vibration.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140803007386901202006-02-24T09:43:00.000-08:002006-02-24T09:43:37.840-08:00Setting the Pin<I><B><P>It is important to "set the pin" correctly in order for the tuning to hold.</B></I> If you look down on the pin, the string comes around the right side of the pin (grands -- it is on the left for uprights) and twirls around it. Therefore if you rotate the pin cw (clockwise), you will tune sharp and vice versa. The string tension is always trying to rotate the pin ccw (counter clock-wise, or flat). Normally, a piano de-tunes flat as you play it. However, because the grip of the pinblock on the pin is so strong, the pin is never straight but is always twisted. </P>
<P>If you rotate it cw and stop, the top of the pin will be twisted cw with respect to the bottom. In this position, the top of the pin wants to rotate ccw (the pin wants to untwist) but can't because it is held by the pinblock. Remember that the string is also trying to rotate it ccw. The two forces together can be sufficient to quickly de-tune the piano flat when you play something loud. </P>
<P>If the pin is turned ccw, the opposite happens -- the pin will want to untwist cw, which opposes the string force. This reduces the net torque on the pin, making the tuning more stable. In fact, you can twist the pin so far ccw that the untwisting force is much larger than the string force and the piano can then de-tune itself sharp as you play. Clearly, you must properly "set the pin" in order produce a stable tuning. This requirement will be taken into account in the following tuning instructions.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140802942833789612006-02-24T09:42:00.000-08:002006-02-24T09:42:22.873-08:00Engaging and Manipulating the Tuning Lever<P>If your tuning lever has adjustable length, pull it out about 3 inches and lock it in place. Hold the handle of the tuning lever in your RH and the socket in your LH and engage the socket over the pin. Orient the handle so that it is approximately perpendicular to the strings and pointing to your right. Lightly jiggle the handle around the pin with your RH and engage the socket with your LH so that the socket is securely engaged, as far down as it will go. From day one, <B><I>develop a habit of jiggling the socket so that it is securely engaged.</B></I> At this point, the handle is probably not perfectly perpendicular to the strings; just choose the socket position so that the handle is as close to perpendicular as the socket position will allow. Now find a way to brace your RH so that you can apply firm pressure on the lever. For example, you can grab the tip of the handle with the thumb and one or two fingers, and brace the arm on the wooden piano frame or brace your pinky against the tuning pins directly under the handle. If the handle is closer to the plate (the metal frame) over the strings, you might brace your hand against the plate. You should not grab the handle like you hold a tennis racket and push-pull to turn the pin -- this will not give enough control. You may be able to do that after years of practice, but in the beginning, grabbing the handle and pushing without bracing against something is too difficult to control accurately. So <B><I>develop a habit of finding good places to brace your hand against, depending on where the handle is</B></I>. Practice these positions making sure that you can exert controlled, constant, powerful pressure on the handle, but do not turn any pins yet. </P>
<P>The lever handle must point to the right so that when you turn it towards you (the string goes sharp), you counteract the force of the string and free the pin from the front side of the hole (towards the string). This allows the pin to turn more freely because of the reduction in friction. When you tune flat, both you and the string are trying to turn the pin in the same direction. Then the pin would turn too easily, except for the fact that both your push and the string's pull jam the pin against the front of the hole, increasing the pressure (friction) and preventing the pin from rotating too easily. If you had placed the handle to the left, you run into trouble for both the sharp and flat motions. For the sharp motion, both you and the string jam the pin against the front of the hole, making it doubly difficult to turn the pin, and damaging the hole. For the flat motion, the lever tends to lift the pin off from the front edge of the hole and reduces the friction. In addition, both the lever and string are turning the pin in the same direction. Now the pin now turns too easily. The lever handle must point to the left for uprights. Looking down on the tuning pin, the lever should point to 3 o'clock for grands and to 9 o'clock for uprights. In both cases, the lever is on the side of the last winding of the string. </P>
<P>Professional tuners do not use these lever positions. Most use 1-2 o'clock for grands and 10-11 o'clock for uprights and Reblitz recommends 6 o'clock for grands and 12 o'clock for uprights. In order to understand why, let's first consider positioning the lever at 12 o'clock on a grand (it is similar at 6 o'clock). Now the friction of the pin with the pinblock is the same for both the sharp and flat motions. However, in the sharp motion, you are going against the string tension and in the flat motion, the string is helping you. Therefore, the difference in force needed between sharp and flat motions is much larger than the difference when the lever is at 3 o'clock, which is a disadvantage. However, unlike the 3 o'clock position, the pin does not rock back and forth during tuning so that when you release the pressure on the tuning lever, the pin does not spring back -- it is more stable -- and you can get higher accuracy. </P>
<P>The 1-2 o'clock position is a good compromise that makes use of both of the advantages of the 3 o'clock and 12 o'clock positions. Beginners do not have the accuracy to take full advantage of the 1-2 o'clock position, so my suggestion is to start with the 3 o'clock position, which should be easier at first, and transition to the 1-2 o'clock position as your accuracy increases. When you become good, the higher accuracy of the 1-2 o'clock position can speed up your tuning so that you can tune each string in just a few seconds. At the 3 o'clock position, you will need to guess how much the pin will spring back and over-tune by that amount, which takes more time. Clearly, exactly where you place the lever will become more important as you improve.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140802687910393682006-02-24T09:37:00.001-08:002006-02-24T09:38:07.916-08:00Pythagorean, Equal, Meantone, and “Well” Temperaments<B><I><P>Historical developments are central to discussions of temperament because the music of the time is tied to the temperament of the time. Pythagoras is credited with inventing the <B>Pythagorean Temperament</B> at around 550 BC, in which the chromatic scale is generated by tuning in perfect 5ths, using the circle of 5ths.</B></I> The twelve perfect fifths in the circle of fifths do not make an exact factor of two. Therefore, the final note you get is not exactly the octave note but is too high in frequency by what is called the "Pythagorean comma", which is about 23 cents (a cent is one hundredths of a semitone). Since a 4th plus a 5th make up an octave, the Pythagorean temperament results in a scale with perfect 4ths and 5ths, except at the end where you get a very bad dissonance. It turns out that tuning in perfect 5ths leaves the 3rds in bad shape. This is another disadvantage of the Pythagorean temperament. Now if we were to tune by contracting each 5th by 23/12 cents, we would end up with exactly one octave and that is one way of tuning an <B>Equal Temperament (ET)</B> scale. In fact, we shall use just such a method in the section on tuning. The ET scale was already known within a hundred years or so after invention of the Pythagorean temperament. Thus ET is not a "modern temperament".</P>
<P>Following the introduction of the Pythagorean temperament, all newer temperaments were efforts at improving on it. The first method was to halve the Pythagorean comma by distributing it among two final 5ths. <B><I>One major development was <B>Meantone Temperament</B>, in which the 3rds were made just instead of the 5ths.</B></I> Musically, 3rds play more prominent roles than 5ths, so that meantone made sense, especially during an age when music made greater use of 3rds. Unfortunately, meantone has a wolf worse than Pythagorean.</P>
<P>The next milestone is represented by Bach's Well Tempered Clavier in which he wrote music for various <B>Well Temperaments (WT)</B>. These were temperaments that struck a compromise between meantone and Pythagorean. This concept worked because Pythagorean tuning ended up with notes that were too sharp, while meantone is too flat. In addition, WT presented the possibility of not only good 3rds, but also good 5ths. <B><I>The simplest WT was devised by Kirnberger, a student of Bach. Its biggest advantage is its simplicity. Better WTs were devised by Werkmeister and by Young. If we broadly classify tunings as Meantone, WT, or Pythagorean, then ET is a WT because ET is neither sharp nor flat.</B></I> There is no record of the temperaments Bach used. We can only guess at the temperaments from the harmonies in his compositions, especially his “Well Tempered Clavier”, and these studies indicate that essentially all the details of tempering were already worked out by Bach’s time (before 1700) and that Bach used a temperament not very different from Werkmeister.</P>
<P>The violin takes advantage of its unique design to circumvent these temperament problems. The open strings make intervals of a 5th with each other, so that the violin naturally tunes Pythagorean. Since the 3rds can always be fingered <B>just</B> (meaning exact), it has all the advantages of the Pythagorean, meantone, and WT, with no wolf in sight! In addition, it has a complete set of frequencies (infinite) within its frequency range. Little wonder that the violin is held in such high esteem by musicians.</P>
<P>In the last 100 years or so, ET had been almost universally accepted because of its musical freedom and the trend towards increasing dissonance. Piano tuners liked it because it can hide minor changes in tuning that can occur just a few days after tuning. All the other temperaments are generically classified as "historical temperaments", which is clearly a misnomer. The historical use of WT gave rise to the concept of key color in which each key, depending on the temperament, endowed specific colors to the music, mainly through the small de-tunings that create "tension" and other effects. This greatly complicated issues because now musicians were dealing not only with pure chords versus wolves, but with colors that were not easily defined. The extent to which the colors can be brought out depends on the piano, the pianist, the listener, and the tuner. Note that the tuner can blend stretch (see "What is stretch?" near the end of section 5) with temperament to control color. After listening to music played on pianos tuned to WT, ET tends to sound more muddy and bland. Thus key color does matter. More important are the wonderful sounds of pure (stretched) intervals in WT. On the other hand, there is always some kind of a wolf in the WTs which is reduced in ET. </P>
<P>For playing most of the music composed around the times of Bach, Mozart, and Beethoven, WT works best. As an example, Beethoven chose chords for the dissonant ninths in the first movement of his Moonlight Sonata that are least dissonant in WT, and are much worse in ET. These great composers were acutely aware of temperament. Most works from Chopin's and Liszt's time were composed with ET in mind and key color is not an issue. Although these compositions sound different in ET and WT to the trained ear, it is not clear that WT is objectionable (for Chopin, etc.) because pure intervals always sound better than detuned ones. The conclusion is that Bach was right: WT should be used for everything, although some musicians might complain that Chopin sounds too bright in WT.</P>
<P>My personal view for the piano is that we should get away from ET because it deprives us of one of the most enjoyable aspects of music -- pure intervals, that was the motivation for creating the chromatic scale. You will see a dramatic demonstration of this if you listen to the last movement of Beethoven's Waldstein played in ET and WT. Meantone can be somewhat extreme unless you are playing music of that period (before Bach), so that we are left with the WTs. For simplicity and ease of tuning, you cannot beat Kirnberger. I believe that once you get used to WT, ET will not sound as good even for Chopin, once you get used to it. Therefore, the world should standardize to the WTs. Which one you choose (Kirnberger, Werckmeister, Valloti, Young) does not make a big difference for most people because those not educated in the temperaments will generally not notice a big difference even among the major temperaments, let alone among the different WTs. This is not to say that we should all use Kirnberger but that we should be educated in the temperaments and have a choice instead of being straight-jacketed into the bland ET. This is not just a matter of taste or even whether the music sounds better. We are talking about developing our musical sensitivity and knowing how to use those really pure intervals.</P>
<P>The biggest disadvantage of WT is that if the piano is out of tune by even a small amount, the dissonance becomes audible, whereas it is much less audible in ET. In fact, most acoustic pianos today will require more frequent tunings if tuned to WT. Therefore, WT will become more practical when the self-tuning pianos become available. There are no such problems with the electronic pianos, and in addition, you can change temperament with the flick of a switch. Another problem with WT is that transposition can change the key color. Of course, WT does not produce all pure intervals – every WT is a compromise just as ET is a compromise.</P>
<P>I believe that these WT drawbacks are minor compared to the advantages; I would be happy if all piano students developed their sensitivity to the point at which they can notice that a piano is very slightly out of tune. And music teachers should be even happier if their students start arguing about which WT is the best. It is about time we listened to Bach, who knew all about ET, but has been trying to tell us to use WT for the last 200 years.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140802654014414052006-02-24T09:37:00.000-08:002006-02-24T09:37:34.016-08:00Temperament, Music, and the Circle of Fifths<P>The above mathematical approach is not the way in which the chromatic scale was historically developed. Musicians first started with chords and tried to find a music scale with the minimum number of notes that would produce those chords. The requirement of a minimum number of notes is obviously desirable since it determines the number of keys, strings, holes, etc. needed to construct a musical instrument. Chords are necessary because if you want to play more than one note at a time, these notes will create dissonances that are unpleasant to the ear unless they form harmonious chords. The reason why dissonances are so unpleasant to the ear may have something to do with the difficulty of processing dissonant information through the brain. It is certainly easier, in terms of memory and comprehension, to deal with harmonious chords than dissonances. Some dissonances are nearly impossible for most brains to figure out if two dissonant notes are played simultaneously. Therefore, if the brain is overloaded with the task of trying to figure out complex dissonances, it becomes impossible to relax and enjoy the music, or follow the musical idea. Clearly, any scale must produce good chords if we are to compose advanced, complex music requiring more than one note at a time.</P>
<B><I><P>We saw above that the optimum number of notes in a scale turned out to be 12. Unfortunately, there isn’t any 12-note scale that can produce exact chords everywhere. Music would sound better if a scale with perfect chords everywhere could be found.</B></I> Many such attempts have been made, mainly by increasing the number of notes per octave, especially using guitars and organs, but none of these scales have gained acceptance. It is relatively easy to increase the number of notes per octave with a guitar-like instrument because all you need to do is to add strings and frets. The latest schemes being devised today involve computer generated scales in which the computer adjusts the frequencies with every transposition; this scheme is called adaptive tuning (Sethares). </P>
<B><I><P>The most basic concept needed to understand temperaments is the concept of the circle of fifths.</B></I> To describe a circle of 5ths, take any octave. Start with the lowest note and go up in 5ths. After two 5ths, you will go outside of this octave. When this happens, go down one octave so that you can keep going up in 5ths and still stay within the original octave. Do this for twelve 5ths, and you will end up at the highest note of the octave! That is, if you start at C4, you will end up with C5 and this is why it is called a circle. Not only that, but every note you hit when playing the 5ths is a different note. This means that the circle of 5ths hits every note once and only once, a key property useful for tuning the scale and for studying it mathematically.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140802602711659862006-02-24T09:32:00.000-08:002006-02-24T09:36:47.926-08:00Mathematics of the Chromatic Scale and Chords<P>Three octaves of the chromatic scale are shown in Table 2.2a using the A, B, C, . . . notation. Black keys on the piano are shown as sharps, e.g. the # on the right of C represents C#, etc., and are shown only for the highest octave. <B><I>Each successive frequency change in the chromatic scale is called a semitone and an octave has 12 semitones.</B></I> The major chords and the integers representing the frequency ratios for those chords are shown above and below the chromatic scale, respectively. The word chord is used here to mean two notes whose frequency ratio is a small integer. Except for multiples of these basic chords, integers larger than about 10 produce chords not readily recognizable to the ear. In reference to Table 2.2a, the most fundamental chord is the octave, in which the frequency of the higher note is twice that of the lower one. The interval between C and G is called a 5th, and the frequencies of C and G are in the ratio of 2 to 3. The major third has four semitones and the minor third has three. <B><I>The number associated with each chord, e.g. four in the 4th, is the number of white keys, inclusive of the two end keys for the C-major scale, and has no further mathematical significance.</B></I> Note that the word "scale" in "chromatic scale", "C-major scale", and "logarithmic or frequency scale" (see below) has different meanings; the second is a subset of the first. </P>
<B><P>TABLE 2.2a: Frequency Ratios of Chords in the Chromatic Scale</P></B>
<TABLE CELLSPACING=0 BORDER=0 WIDTH=600>
<TR><TD WIDTH="19%" VALIGN="MIDDLE">
<P>|--Octave--<BR>
<B>CDEFGAB<BR>
1</B></TD>
<TD WIDTH="14%" VALIGN="MIDDLE">
<P>|--5th--<BR>
<B>C D E F<BR>
2</B></TD>
<TD WIDTH="13%" VALIGN="MIDDLE">
<P>|--4th--<BR>
<B>G A B<BR>
3</B></TD>
<TD WIDTH="16%" VALIGN="MIDDLE">
<P>|-Maj.3rd-<BR>
<B>C # D #<BR>
4</B></TD>
<TD WIDTH="16%" VALIGN="MIDDLE">
<P>|-Min.3rd-<BR>
<B>E F #<BR>
5</B></TD>
<TD WIDTH="16%" VALIGN="MIDDLE">
<B><P>|<BR>
G # A # B<BR>
6</B></TD>
<TD WIDTH="5%" VALIGN="MIDDLE">
<B><P>|<BR>
C<BR>
8</B></TD>
</TR>
</TABLE>
<P>We can see from the above that a 4th and a 5th "add up" to an octave and a major 3rd and a minor 3rd "add up" to a 5th. Note that this is an addition in logarithmic space, as explained below. The missing integer 7 is also explained below.</P>
<B><I><P>The "equal tempered" (ET) chromatic scale consists of "equal" half-tone or semitone rises for each successive note.</B></I> They are equal in the sense that the ratio of the frequencies of any two adjacent notes is always the same. This property ensures that each note is the same as any other note (except for pitch). This uniformity of the notes allows the composer or performer to use any key without hitting bad dissonances, as further explained below. There are 12 equal semitones in an octave of an ET scale and each octave is an exact factor of two in frequency. Therefore, the frequency change for each semitone is given by</P>
<P>&#9;semitone<SUP>12</SUP>&#9;= 2&#9;or<BR>
&#9;semitone&#9;= 2<SUP>1/12</SUP> = 1.05946. . . . . . . . . . . . . . . . . . Eq. (2.1)</P>
<P>Eq. (2.1) defines the ET chromatic scale and allows the calculation of the frequency ratios of "chords" in this scale. How do the "chords" in ET compare with the frequency ratios of the ideal chords? <B><I>The comparisons are shown in Table 2.2b and demonstrate that the chords from the ET scale are extremely close to the ideal chords.</P>
</I><P>TABLE 2.2b: Ideal Chords versus the Equal Tempered Scale</P></B>
<TABLE CELLSPACING=0 BORDER=0 CELLPADDING=7 WIDTH=700>
<TR><TD WIDTH="17%" VALIGN="TOP">
<B><U><P>Chord</B></U></TD>
<TD WIDTH="24%" VALIGN="TOP">
<B><U><P>Freq. Ratio</B></U></TD>
<TD WIDTH="36%" VALIGN="TOP">
<B><U><P>Eq. Tempered Scale</B></U></TD>
<TD WIDTH="23%" VALIGN="TOP">
<B><U><P>Difference</B></U></TD>
</TR>
<TR><TD WIDTH="17%" VALIGN="TOP">
<P>Min.3rd:</TD>
<TD WIDTH="24%" VALIGN="TOP">
<P>6/5 = 1.2000</TD>
<TD WIDTH="36%" VALIGN="TOP">
<P>semitone<SUP>3</SUP> = 1.1892</TD>
<TD WIDTH="23%" VALIGN="TOP">
<P>+0.0108</TD>
</TR>
<TR><TD WIDTH="17%" VALIGN="TOP">
<P>Maj.3rd:</TD>
<TD WIDTH="24%" VALIGN="TOP">
<P>5/4 = 1.2500</TD>
<TD WIDTH="36%" VALIGN="TOP">
<P>semitone<SUP>4</SUP> = 1.2599</TD>
<TD WIDTH="23%" VALIGN="TOP">
<P>-0.0099</TD>
</TR>
<TR><TD WIDTH="17%" VALIGN="TOP">
<P>Fourth:</TD>
<TD WIDTH="24%" VALIGN="TOP">
<P>4/3 = 1.3333</TD>
<TD WIDTH="36%" VALIGN="TOP">
<P>semitone<SUP>5</SUP> = 1.3348</TD>
<TD WIDTH="23%" VALIGN="TOP">
<P>-0.0015</TD>
</TR>
<TR><TD WIDTH="17%" VALIGN="TOP">
<P>Fifth:</TD>
<TD WIDTH="24%" VALIGN="TOP">
<P>3/2 = 1.5000</TD>
<TD WIDTH="36%" VALIGN="TOP">
<P>semitone<SUP>7</SUP> = 1.4983</TD>
<TD WIDTH="23%" VALIGN="TOP">
<P>+0.0017</TD>
</TR>
<TR><TD WIDTH="17%" VALIGN="TOP">
<P>Octave:</TD>
<TD WIDTH="24%" VALIGN="TOP">
<P>2/1 = 2.0000</TD>
<TD WIDTH="36%" VALIGN="TOP">
<P>semitone<SUP>12</SUP> = 2.0000</TD>
<TD WIDTH="23%" VALIGN="TOP">
<P>0.0000</TD>
</TR>
</TABLE>
<B><I><P>The errors for the 3rds are the worst, over five times the errors in the other chords, but are still only about 1%.</B></I> Nonetheless, these errors are readily audible, and some piano aficionados have generously dubbed them "the rolling thirds" while in reality, they are unacceptable dissonances. It is a defect that we must learn to live with, if we are to adopt the ET scale. The errors in the 4ths and 5ths produce beats of about 1 Hz near middle C, which is barely audible in most pieces of music; however, this beat frequency doubles for every higher octave. </P>
<P>The integer 7, if it were included in Table 2.2a, would have represented a chord with the ratio 7/6 and would correspond to a semitone squared. The error between 7/6 and a semitone squared is over 4% and is too large to make a musically acceptable chord and was therefore excluded from Table 2.2a. It is just a mathematical accident that the 12-note chromatic scale produces so many ratios close to the ideal chords. <B><I>Only the number 7, out of the smallest 8 integers, results in a totally unacceptable chord.</B></I> <B><I>The chromatic scale is based on a lucky mathematical accident in nature! It is constructed by using the smallest number of notes that gives the maximum number of chords.</B></I> No wonder early civilizations believed that there was something mystical about this scale. Increasing the number of keys in an octave does not result in much improvement of the chords until the numbers become quite large, making that approach impractical for most musical instruments.</P>
<P>Note that the frequency ratios of the 4th and 5th do not add up to that of the octave (1.5000 + 1.3333 = 2.8333 vs 2.0000). Instead, they add up in logarithmic space because (3/2)x(4/3) = 2. In logarithmic space, multiplication becomes addition. Why might this be significant? The answer is because the geometry of the cochlea of the ear seems to have a logarithmic component. Detecting acoustic frequencies on a logarithmic scale accomplishes two things: you can hear a wider frequency range for a given size of cochlea, and analyzing ratios of frequencies becomes simple because instead of dividing or multiplying two frequencies, you only need to subtract or add their logarithms. For example, if C3 is detected by the cochlea at one position and C4 at another position 2mm away, then C5 will be detected at a distance of 4 mm, exactly as in the slide rule calculator. To show you how useful this is, given F5, the brain knows that F4 will be found 2mm back! Therefore, chords (remember, chords are frequency divisions) are particularly simple to analyze in a logarithmically constructed cochlea. </B></I>When we play chords, we are performing mathematical operations in logarithmic space on a mechanical computer called the piano, as was done in the 1950's with the slide rule.<B><I> Thus the logarithmic nature of the chromatic scale has many more consequences than just providing a wider frequency range.</B></I> The logarithmic scale assures that the two notes of every chord are separated by the same distance no matter where you are on the piano. By adopting a logarithmic chromatic scale, the piano keyboard is mathematically matched to the human ear in a mechanical way! This is probably one reason for why harmonies are pleasant to the ear - harmonies are most easily deciphered and remembered by the human hearing mechanism.</P>
<P>Suppose that we did not know Eq. 2.1; can we generate the ET chromatic scale from the chord relationships? If the answer is yes, a piano tuner can tune a piano without having to make any calculations. These chord relationships, it turns out, completely determine the frequencies of all the notes of the 12 note chromatic scale. A temperament is a set of chord relationships that provides this determination. From a musical point of view, there is no single "chromatic scale" that is best above all else although ET has the unique property that it allows free transpositions. Needless to say, <B><I>ET is not the only musically useful temperament, and we will discuss other temperaments below.</B></I> Temperament is not an option but a necessity; we <I>must</I> choose a temperament in order to accommodate these mathematical difficulties. <B><I>No musical instrument based on the chromatic scale is completely free of temperament.</B></I> For example, the holes in wind instruments and the frets of the guitar must be spaced for a specific tempered scale. The violin is a devilishly clever instrument because it avoids all temperament problems by spacing the open strings in fifths. If you tune the A(440) string correctly and tune all the others in 5ths, these others will be close, but not tempered. You can still avoid temperament problems by fingering all notes except one (usually A-440). In addition, the vibrato is larger than the temperament corrections, making temperament differences inaudible.</P>
<B><I><P>The requirement of tempering arises because a chromatic scale tuned to one scale (e.g., C-major with perfect chords) does not produce acceptable chords in other scales.</I></B> If you wrote a composition in C-major having many perfect chords and then transposed it, terrible dissonances can result. There is an even more fundamental problem. Perfect chords in one scale also produce dissonances in other scales needed in the same piece of music. Tempering schemes were therefore devised to minimize these dissonances by minimizing the de-tuning from perfect chords in the most important chords and shifting most of the dissonances into the less used chords. The dissonance associated with the worst chord came to be known as <B>“the wolf”</B>.</P>
<P>The main problem is, of course, chord purity; the above discussion makes it clear that no matter what you do, there is going to be a dissonance somewhere. <B><I>It might come as a shock to some that the piano is a fundamentally imperfect instrument!</B></I> We are left to deal forever with some compromised chords in almost every scale. </P>
<P>The name "chromatic scale" generally applies to any 12-note scale with any temperament. Naturally, the chromatic scale of the piano does not allow the use of frequencies between the notes (as you can with the violin), so that there is an infinite number of missing notes. In this sense, the chromatic scale is incomplete. Nonetheless, the 12-note scale is sufficiently complete for the majority of musical applications. The situation is analogous to digital photography. When the resolution is sufficient, you cannot see the difference between a digital photo and an analog one with much higher information density. Similarly, <B><I>the 12-note scale apparently has sufficient pitch resolution for a sufficiently large number of musical applications.</B></I> This 12-note scale is a good compromise between having more notes per octave for greater completeness and having enough frequency range to span the range of the human ear, for a given instrument or musical notation system with a limited number of notes. </P>
<P>There is healthy debate about which temperament is best musically. ET was known from the earliest history of tuning. There are definite advantages to standardizing to one temperament, but that is probably not possible or even desirable in view of the diversity of opinions on music and the fact that much music now exist, that were written with particular temperaments in mind. Therefore we shall now explore the various temperaments.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140800804612128232006-02-24T09:06:00.000-08:002006-02-24T09:06:44.620-08:00Beethoven (5th Symphony, Appassionata, Waldstein)<P>The use of mathematical devices is deeply embedded in Beethoven's music. Therefore, this is one of the best places to dig for information on the relationship between mathematics and music. I'm not saying that other composers do not use mathematical devices. Practically every musical composition has mathematical underpinnings. However, Beethoven was able to extend these mathematical devices to the extreme. It is by analyzing these extreme cases that we can find more convincing evidence on what types of devices he used.</P>
<P>We all know that Beethoven never really studied advanced mathematics. Yet he incorporates a surprising amount of math in his music, at very high levels. The beginning of his Fifth Symphony is a prime case, but examples such as this are legion. He "used" group theory type concepts to compose this famous symphony. In fact, he used what crystallographers call the Space Group of symmetry transformations! This Group governs many advanced technologies, such as quantum mechanics, nuclear physics, and crystallography that are the foundations of today's technological revolution. At this level of abstraction, <I>a crystal of diamond and Beethoven's 5th symphony are one and the same!</I> I will explain this remarkable observation below.</P>
<P>The Space Group that Beethoven "used" (he certainly had a different name for it) has been applied to characterize crystals, such as silicon and diamond. It is the properties of the Space Group that allow crystals to grow defect free and therefore, the Space Group is the very basis for the existence of crystals. Since crystals are characterized by the Space Group, an understanding of the Space Group provides a basic understanding of crystals. This was neat for materials scientists working to solve communications problems because the Space Group provided the framework from which to launch their studies. It's like the physicists needed to drive from New York to San Francisco and the mathematicians handed them a map! That is how we perfected the silicon transistor, which led to integrated circuits and the computer revolution. So, what is the Space Group? And why was this Group so useful for composing this symphony?</P>
<P>Groups are defined by a set of properties. Mathematicians found that groups defined in this way can be mathematically manipulated and physicists found them to be useful: that is, these particular groups that interested mathematicians and scientists provide us with a pathway to reality. One of the properties of groups is that they consist of Members and Operations. Another property is that if you perform an Operation on a Member, you get another Member of the same Group. A familiar group is the group of integers: -1, 0, 1, 2, 3, etc. An Operation for this group is addition: 2 + 3 = 5. Note that the application of the operation + to Members 2 and 3 yields another Member of the group, 5. Since Operations transform one member into another, they are also called Transformations. A Member of the Space Group can be anything in any space: an atom, a frog, or a note in any musical space dimension such as pitch, speed, or loudness. The Operations of the Space Group relevant to crystallography are Translation, Rotation, Mirror, Inversion, and the Unitary operation. These are almost self explanatory (Translation means you move the Member some distance in that space) except for the Unitary operation which basically leaves the Member unchanged. However, it is somewhat subtle because it is not the same as the equality transformation, and is therefore always listed last in textbooks. Unitary operations are generally associated with the most special member of the group, which we might call the Unitary Member. In the integer group noted above, this Member would be 0 for addition and 1 for multiplication (5+0 = 5x1 = 5).</P>
<P>Let me demonstrate how you might use this Space Group, in ordinary everyday life. Can you explain why, when you look into a mirror, the left hand goes around to the right (and vice versa), but your head doesn't rotate down to your feet? The Space Group tells us that you can't rotate the right hand and get a left hand because left-right is a mirror operation, not a rotation. Note that this is a strange transformation: your right hand becomes your left hand in the mirror; therefore, the wart on your right hand will be on your left hand image in the mirror. This can become confusing for a symmetric object such as a face because a wart on one side of the face will look strangely out of place in a photograph, compared to your familiar image in a mirror. The mirror operation is why, when you look into a flat mirror, the right hand becomes a left hand; however, a mirror cannot perform a rotation, so your head stays up and the feet stay down. Curved mirrors that play optical tricks (such as reversing the positions of the head and feet) are more complex mirrors that can perform additional Space Group operations, and group theory will be just as helpful in analyzing images in a curved mirror. The solution to the flat mirror image problem appeared to be rather easy because we had a mirror to help us, and we are so familiar with mirrors. The same problem can be restated in a different way, and it immediately becomes much more difficult, so that the need for group theory to help solve the problem becomes more obvious. If you turned a right hand glove inside out, will it stay right hand or will it become a left hand glove? I will leave it to you to figure that one out (hint: use a mirror).</P>
<P>Let's see how Beethoven used his intuitive understanding of spatial symmetry to compose his 5th Symphony. That famous first movement is constructed largely by using a single short musical theme consisting of four notes, of which the first three are repetitions of the same note. Since the fourth note is different, it is called the surprise note, and carries the beat. This musical theme can be represented schematically by the sequence 555<B>3</B>, where <B>3</B> is the surprise note. This is a pitch based space group; Beethoven used a space with 3 dimensions, pitch, time, and volume. I will consider only the pitch and time dimensions in the following discussions. Beethoven starts his Fifth Symphony by first introducing a Member of his Group: three repeat notes and a surprise note, 555<B>3</B>. After a momentary pause to give us time to recognize his Member, he performs a Translation operation: 444<B>2</B>. Every note is translated down. The result is another Member of the same Group. After another pause so that we can recognize his Translation operator, he says, "Isn't this interesting? Let's have fun!" and demonstrates the potential of this Operator with a series of translations that creates music. In order to make sure that we understand his construct, he does not mix other, more complicated, operators at this time. In the ensuing series of bars, he then successively incorporates the Rotation operator, creating <B>3</B>555, and the Mirror operator, creating <B>7</B>555. Somewhere near the middle of the 1st movement, he finally introduces what might be interpreted as the Unitary Member: 555<B>5</B>. Note that these groups of 5 identical notes are simply repeated, which is the Unitary operation.</P>
<P>In the final fast movements, he returns to the same group, but uses only the Unitary Member, and in a way that is one level more complex. It is always repeated three times. What is curious is that this is followed by a fourth sequence -- a surprise sequence 765<B>4</B>, which is not a Member. Together with the thrice repeated Unitary Member, the surprise sequence forms a Supergroup of the original Group. He has generalized his Group concept! The supergroup now consists of three members and a non-member of the initial group, which satisfies the conditions of the initial group (three repeats and a surprise).</P>
<P>Thus, the beginning of Beethoven's Fifth symphony, when translated into mathematical language, reads just like the first chapter of a textbook on group theory, almost sentence for sentence! Remember, group theory is one of the highest forms of mathematics. The material is even presented in the correct order as they appear in textbooks, from the introduction of the Member to the use of the Operators, starting with the simplest, Translation, and ending with the most subtle, the Unitary operator. He even demonstrates the generality of the concept by creating a supergroup from the original group. </P>
<P>Beethoven was particularly fond of this four-note theme, and used it in many of his compositions, such as the first movement of the Appassionata piano sonata, see bar 10, LH. Being the master that he is, he carefully avoids the pitch based Space Group for the Appassionata and uses different spaces -- he transforms them in tempo space and volume space (bars 234 to 238). This is further support for the idea that he must have had an intuitive grasp of group theory and consciously distinguished between these spaces. It seems to be a mathematical impossibility that this many agreements of his constructs with group theory just happened by accident, and is virtual proof that he was somehow playing around with these concepts.</P>
<P>Why was this construct so useful in this introduction? It certainly provides a uniform platform on which to hang his music. The simplicity and uniformity allow the audience to concentrate only on the music without distraction. It also has an addictive effect. These subliminal repetitions (the audience is not supposed to know that he used this particular device) can produce a large emotional effect. It is like a magician's trick -- it has a much larger effect if we do not know how the magician does it. It is a way of controlling the audience without their knowledge. Just as Beethoven had an intuitive understanding of this group type concept, we may all feel that some kind of pattern exists, without recognizing it explicitly. Mozart accomplished a similar effect using repetitions.</P>
<P>Knowledge of these group type devices that he uses is very useful for playing his music, because it tells you exactly what you should and should not do. Another example of this can be found in the 3rd movement of his Waldstein sonata, where the entire movement is based on a 3-note theme represented by 15<B>5 </B>(the first CG<B>G </B>at the beginning). He does the same thing with the initial arpeggio of the 1st movement of the Appassionata, with a theme represented by 53<B>1</B> (the first CAb<B>F</B>). In both cases, unless you maintain the beat on the last note, the music loses its structure, depth and excitement. This is particularly interesting in the Appassionata, because in an arpeggio, you normally place the beat on the first note, and many students actually make that mistake. As in the Waldstein, this initial theme is repeated throughout the movement and is made increasingly obvious as the movement progresses. But by then, the audience is addicted to it and does not even notice that it is dominating the music. For those interested, you might look near the end of the 1st movement of the Appassionata where he transforms the theme to 31<B>5</B> and raises it to an extreme and almost ridiculous level at bar 240. Yet most in the audience will have no idea what device Beethoven was using, except to enjoy the wild climax, which is obviously ridiculously extreme, but by now carries a mysterious familiarity because the construct is the same, and you have heard it hundreds of times. Note that this climax loses much of its effect if the pianist does not bring out the theme (introduced in the first bar!) and emphasize the beat note. </P>
<P>Beethoven tells us the reason for the inexplicable 53<B>1</B> arpeggio in the beginning of the Appassionata when the arpeggio morphs into the main theme of the movement at bar 35. That is when we discover that the arpeggio at the beginning is an inverted and schematized form of his main theme, and why the beat is where it is. Thus the beginning of this piece, up to bar 35, is a psychological preparation for one of the most beautiful themes he composed. He wanted to implant the idea of the theme in our brain before we heard it! That may be one explanation for why this strange arpeggio is repeated twice at the beginning using an illogical chord progression. With analysis of this type, the structure of the entire 1st movement becomes apparent, which helps us to memorize, interpret, and play the piece correctly.</P>
<P>The use of group theoretical type concepts might be just an extra dimension that Beethoven wove into his music, perhaps to let us know how smart he was, in case we still didn't get the message. It may or may not be the mechanism with which he generated the music. Therefore, the above analysis gives us only a small glimpse into the mental processes that inspire music. Simply using these devices does not result in music. Or, are we coming close to something that Beethoven knew but didn't tell anyone?</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140800768140533022006-02-24T09:05:00.000-08:002006-02-24T09:06:08.146-08:00Mozart (Eine Kleine Nachtmusik, Sonata K300)<P>When I first heard of Mozart's formula, I felt a great excitement, because I thought that it might shed light on music theory and on music itself. You may at first be disappointed, as I was, when you find out that Mozart's formula, as we know it today, appears to be strictly structural. Structural analyses have so far not yet provided much information on how you come up with famous melodies; but then, music theory doesn't either. Today's music theory only helps to compose "correct" music or expand on it once you have come up with a musical idea. Music theory is a classification of families of notes and their arrangements in certain patterns. We can not yet rule out the possibility that music is ultimately based on certain identifiable types of structural patterns. I first learned of Mozart's formula at a lecture given by a music professor. I have since lost the reference -- if anyone reading this book knows of a reference (professor’s name, his institution), please let me know.</P>
<P>It is now known that Mozart composed practically all of his music, from when he was very young, according to a single formula that expanded his music by over a factor of ten. That is, whenever he concocted a new melody that lasted one minute, he knew that his final composition would be at least ten minutes long. Sometimes, it was a <I>lot</I> longer. The first part of his formula was to repeat every theme. These themes were generally very short -- only 4 to 10 notes, much shorter than you would think when you think of a musical theme. These themes, that are much shorter than the over-all melody, simply disappear into the melody because they are too short to be recognized. This is why we do not normally notice them, and is almost certainly a conscious construct by the composer. The theme would then be modified two or three times and repeated again to produce what the audience would perceive as a continuous melody. These modifications consisted of the use of various mathematical and musical symmetries such as inversions, reversals, harmonic changes, clever positioning of ornaments, etc. These repetitions would be assembled to form a section and the whole section would be repeated. The first repetition provides a factor of two, the various modifications provide another factor of two to six (or more), and the final repetition of the entire section provides another factor of two, or 2x2x2 = 8 at a minimum. In this way, he was able to write huge compositions with a minimum of thematic material. In addition, his modifications of the original theme followed a particular order so that certain moods or colors of music were arranged in the same order in each composition. </P>
<P>Because of this pre-ordained structure, he was able to write down his compositions from anywhere in the middle, or one voice at a time, since he knew ahead of time where each part belonged. And he did not have to write down the whole thing until the last piece of the puzzle was in place. He could also compose several pieces simultaneously, because they all had the same structure. This formula made him look like more of a genius than he really was. This naturally leads us to question: how much of his reputed "genius" was simply an illusion of such machinations? This is not to question his genius -- the music takes care of that! However, many of the wonderful things that these geniuses did were the result of relatively simple devices and we can all take advantage of that by finding out the details of these devices. For example, knowing Mozart's formula makes it easier to dissect and memorize his compositions. The first step towards understanding his formula is to be able to analyze his repetitions. They are not simple repetitions; Mozart used his genius to modify and disguise the repetitions so that they produced music and, more importantly, so that the fact of the repetition will not be recognized.</P>
<P>As an example of repetitions, let's examine the famous melody in the Allegro of his Eine Kleine Nachtmusik. This is the melody that Salieri played and the pastor recognized in the beginning of the movie, "Amadeus". That melody is a repetition posed as a question and an answer. The question is a male voice asking, "Hey, are you coming?" And the reply is a female voice, "OK, OK, I'm coming!" The male statement is made using only two notes, a commanding fourth apart, repeated three times (six notes), and the question is created by adding three rising notes at the end (this appears to be universal among most languages -- questions are posed by raising the voice at the end). Thus the first part consists of 9 notes. The repetition is an answer in a female voice because the pitch is higher, and is again two notes, this time a sweeter minor third apart, repeated (you guessed it!) three times (six notes). It is an answer because the last three notes wiggle down. Again, the total is 9 notes. The efficiency with which he created this construct is amazing. What is even more incredible is how he disguises the repetition so that when you listen to the whole thing, you would not think of it as a repetition. Practically all of his music can be analyzed in this way. If you are not yet convinced, take any of his music and analyze it, and you will find this pattern. </P>
<P>Let's look at another example, the Sonata #16 in A, K300 (or KV331, the one with the Alla Turca ending). The basic unit of the beginning theme is a quarter note followed by an eighth note. The first introduction of this unit in bar 1 is disguised by the addition of the 16th note. This introduction is followed by the basic unit, completing bar 1. Thus in the first bar, the unit is repeated twice. He then translates the whole double unit of the 1st bar down in pitch and creates bar 2. The 3rd bar is just the basic unit repeated twice. In the 4th bar, he again disguises the first unit by use of the 16th notes. Bars 1 to 4 are then repeated with minor modifications in bars 5-8. From a structural viewpoint, every one of the first 8 bars is patterned after the 1st bar. From a melodic point of view, these 8 bars produce two long melodies with similar beginnings but different endings. Since the whole 8 bars is repeated, he has basically multiplied his initial idea embodied in the 1st bar by 16! If you think in terms of the basic unit, he has multiplied it by 32. But then he goes on to take this basic unit and creates incredible variations to produce the first part of the sonata, so the final multiplication factor is even larger. He uses repetitions of repetitions. By stringing the repetitions of modified units, he creates a final melody that sounds like a long melody until you break it up into its components. </P>
<P>In the 2nd half of this exposition, he introduces new modifications to the basic unit. In bar 10, he first adds an ornament with melodic value to disguise the repetition and then introduces another modification by playing the basic unit as a triplet. Once the triplet is introduced, it is repeated twice in bar 11. Bar 12 is similar to bar 4; it is a repetition of the basic unit, but structured in such a way as to act as a conjunction between the preceding 3 related bars and the following 3 related bars. Thus bars 9 to 16 are similar to bars 1 to 8, but with a different musical idea. The final 2 bars (17 and 18) provide the ending to the exposition. With these analyses as examples, you should now be able to dissect the remainder of this piece. You will find that the same pattern of repetitions is found throughout the entire piece. As you analyze more of his music you will need to include more complexities; he may repeat 3 or even 4 times, and mix in other modifications to hide the repetitions. What is clear is that he is a master of disguise so that the repetitions and other structures are not usually obvious when you just listen to the music without any intent to analyze it.</P>
<P>Mozart's formula certainly increased his productivity. Yet he may have found certain magical (hypnotic? addictive?) powers to repetitions of repetitions and he probably had his own musical reasons for arranging the moods of his themes in the sequence that he used. That is, if you further classify his melodies according to the moods they evoke, it is found that he always arranged the moods in the same order. The question here is, if we dig deeper and deeper, will we just find more of these simple structural/mathematical devices, just stacked one on top of each other, or is there more to music? Almost certainly, there must be more, but no one has yet put a finger on it, not even the great composers themselves -- at least, as far as they have told us. Thus it appears that the only thing we mortals can do is to keep digging.</P>
<P>The music professor mentioned above who lectured on Mozart’s formula also stated that the formula is followed so strictly that it can be used to identify Mozart’s compositions. However, elements of this formula were well known among composers. Thus Mozart is not the inventor of this formula and similar formulas were used widely by composers of his time. Some of Salieri’s compositions follow a very similar formula; perhaps this was an attempt by Salieri so emulate Mozart. Thus you will need to know some details of Mozart’s specific formula in order to use it to identify his compositions.</P>
<P>There is little doubt that a strong interplay exists between music and genius. We don't even know if Mozart was a composer because he was a genius or if his extensive exposure to music from birth created the genius. The music doubtless contributed to his brain development. It may very well be that the best example of the "Mozart effect" was Wolfgang Amadeus himself, even though he did not have the benefit of his own masterpieces. Today, we are just beginning to understand some of the secrets of how the brain works. For example, until recently, we had it partly wrong when we thought that certain populations of mentally handicapped people had unusual musical talent. It turns out that music has a powerful effect on the actual functioning of the brain and its motor control. This is one of the reasons why we always use music when dancing or exercising. The best evidence for this comes from Alzheimer's patients who have lost their ability to dress themselves because they cannot recognize each different type of clothing. It was discovered that when this procedure is set to the proper music, these patients can often dress themselves! "Proper music" is usually music that they heard in early youth or their favorite music. Thus mentally handicapped people who are extremely clumsy when performing daily chores can suddenly sit down and play the piano if the music is the right type that stimulates their brain. Therefore, they may not be musically talented; instead, it is the music that is giving them new capabilities. It is not only music that has these magical effects on the brain, as evidenced by savants who can memorize incredible amounts of information or carry out mathematical feats normal folks cannot perform. There is a more basic internal rhythm in the brain that music happens to excite. Therefore, these savants may not be talented but are just using some of the methods of this book, such as mental play. Just as good memorizers have brains that are automatically memorizing everything they encounter, some savants may be repeating music or mathematical thoughts in their heads all the time, which would explain why they cannot perform ordinary chores – because their brains are already preoccupied with something else.</P>
<P>If music can produce such profound effects on the handicapped, imagine what it could do to the brain of a budding genius, especially during the brain's development in early childhood. These effects apply to anyone who plays the piano, not just the handicapped or the genius.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140800718762073512006-02-24T09:04:00.000-08:002006-02-24T09:05:18.776-08:00Mozart's Formula, Beethoven and Group TheoryThere is an intimate, if not absolutely essential, relationship between mathematics and music. At the very least, they share a large number of the most fundamental properties in common, starting with the fact that the chromatic scale is a simple logarithmic equation (see Chapt. Two, Section 2) and that the basic chords are ratios of the smallest integers. Now few musicians are interested in mathematics for mathematics' sake. However, practically everyone is curious and has wondered at one time or other whether mathematics is somehow involved in the creation of music. Is there some deep, underlying principle that governs both math and music? In addition, there is the established fact that every time we succeeded in applying mathematics to a field, we have made tremendous strides in advancing that field. One way to start investigating this relationship is to study the works of the greatest composers from a mathematical point of view. The following analyses contain no inputs from music theory.Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140798186000614042006-02-24T08:13:00.000-08:002006-02-24T08:25:13.023-08:00Future Research Topics<P>Every subsection in this section is incomplete; I am just putting down some initial ideas.</P>
<P>This book is based on the scientific approach, which ensures that errors are corrected as soon as possible, that all known facts are explained, documented, and organized in a useful way, and that we only make forward progress. The past situation of one piano teacher teaching a very useful method and another knowing nothing about it, or two teachers teaching completely opposite methods, should not occur. An important part of the scientific approach is a discussion of what is still unknown and what still needs to be researched. The following is a collection of such topics.</P>
<LI><P><B>Momentum Theory of Piano Playing</B></P></LI>
<P>Slow play in piano is called “playing in the static limit”. This means that when depressing a key, the force of the finger coming down is the main force used in the playing. As we speed up, we transition from the static limit to the momentum limit. This means that the momenta of the hand, arms, fingers, etc., play much more important roles than the force in depressing the keys. Of course, force is needed to depress the key, but when in the momentum limit, the force and motion are usually 180 degrees out of phase. That is, your finger is moving up when your finger muscles are trying to press it down! This happens at high speed because you had earlier lifted the finger so rapidly that you have to start depressing it on its way up so that you can reverse its action for the next strike. The actual motions a complex because you use the hand, arms, and body to impart and absorb the momenta. This is one of the reasons why the entire body gets involved in the playing, especially when playing fast. Note that the swing of the pendulum and the dribbling of the basketball are in the momentum limit. In piano playing, you are generally somewhere between the static and momentum limits with increasing tendency towards momentum limit with increasing speed.</P>
<P>In static play, the force vector and motion of the finger are in phase. As we transition to momentum play, a phase difference develops, until, in the pure momentum regime, the phase difference is 180 degrees, as it is in the pendulum.</P>
<P>The importance of momentum play is obvious; it involves many new finger/hand motions that are not possible in static play. Thus knowing which motions are of the static or momentum type will go a long way toward understanding how to execute them and when to use them. Because momentum play has never been discussed in the literature until now, there is a vast area of piano play for which we have very little understanding.</P>
<LI><P><B>The Physiology of Technique</B></P></LI>
<P>We still lack even a rudimentary understanding of the bio-mechanical processes that underlie technique. It certainly originates in the brain, and is probably associated with how the nerves communicate with the muscles, especially the rapid muscles. What are the biological changes that accompany technique? when fingers are “warmed up”?</P>
<LI><P><B>Brain Research (HS vs HT Play, etc.)</B></P></LI>
<P>Brain research will be one of the most important fields of medical research in the near future. This research will initially concentrate on preventing mental deterioration with age (such as curing Alzheimer’s disease). Then concurrent efforts at actually controlling the growth of mental capabilities will surely develop. Music should play an important role in such developments because we can communicate aurally with infants long before any other method, and it is already clear that, the earlier you start the control process, the better the results.</P>
<P>We are all familiar with the fact that, even if we can play HS quite well, HT may still be difficult. Why is HT so much more difficult? One of the reasons may be that the two hands are controlled by the different halves of the brain. If so, then learning HT requires the brain to develop ways to coordinate the two halves. This would mean that HS and HT practice use completely different types of brain functions and supports the contention that these skills should be developed separately so that we work on one skill at a time. One intriguing possibility is that we may be able to develop HT parallel sets that can solve this problem.</P>
<LI><P><B>What Causes Nervousness?</B></P></LI>
<P>In piano pedagogy, nervousness has been “swept under the rug” (ignored) for too long. We need to study it from medical and physiological points of view. We need to know if some individuals can benefit from proper medication. Moreover, is there a medical or psychological regiment whereby it can eventually be eliminated? Finally, from a formal psychological point of view, we must develop a teaching procedure that will reduce nervousness. Nervousness is clearly the result of a mental attitude/reaction/perception, and is therefore very amenable to active control. For example, pianists who play popular/jazz type music seem to be much less nervous in general than those who play classical. There is no reason why we shouldn’t investigate why this is so, and take advantage of this phenomenon.</P>
<LI><P><B>Causes and Remedies for Tinnitus</B></P></LI>
<P>Cochlea structure, high and low frequency tinnitus.</P>
<P>There is evidence that moderate intake of aspirin can slow down hearing loss with age. However, there is also evidence that aspirin, under certain conditions, can aggravate tinnitus. There does not appear to be any evidence that tinnitus is caused purely by age; instead, there is ample evidence that it is caused by infection, disease, accidents, and abuse of the ear. Therefore, in most of these cases, the causes and the types of damage can be directly studied.</P>
<LI><P><B>What is Music?</B></P></LI>
<P>Cochlea structure vs music scales and chords. Parameters: timing (rhythm), pitch, patterns (language, emotions), volume, speed. Musical information processing in brain.</P>
<LI><P><B>At What Age to Start Piano?</B></P></LI>
<P>We need medical/psychological/sociological studies into how/when we should start youngsters. This type of research is already starting to be conducted in sports, at least informally, by individual sports organizations that have developed methods for teaching youngsters down to about 2 years of age. In music, we can start as soon as the babies are born, by letting them listen to the appropriate types of music. In music, we are probably interested more in the development of the brain than in acquiring motor skills. Because we expect brain research to explode in the near future, this is an opportune time to take advantage of that research and use the results for learning piano.</P>
<LI><P><B>The Future of Piano</B></P></LI>
<P>Finally, we look into the future. The “Testimonials” section gives ample evidence that our new approach to piano practice will enable practically anyone to learn piano to her/is satisfaction. It will certainly increase the number of pianists. Therefore, the following questions become very important: (1) can we calculate the expected increase in pianists? (2) what will this increase do to the economics of the piano: performers, teachers, technicians, and manufacturers, and (3) if piano popularity explodes, what will be the main motivation for such large numbers of people to learn piano?</P>
<P>Piano teachers will agree than 90% of piano students never really learn piano in the sense that they will not be able to play to their satisfaction and basically give up trying to become accomplished pianists. Since this is a well known phenomenon, it discourages youngsters and their parents from deciding to start piano lessons. Since serious involvement with piano will interfere materially with the business of making a living, the economic factor also discourages entry into piano. There are many more negative factors that limit the popularity of the piano (lack of good teachers, high expense of good pianos and their maintenance, etc.), almost all of which are eventually related to the fact that piano has been so difficult to learn. Probably only 10% of those who might have tried piano ever decide to give it a try. Therefore, we can reasonably expect the popularity of the piano to increase by 100 times if the promise of this book can be fulfilled.</P>
<P>Such an increase would mean that a large fraction of the population in developed countries would learn piano. Since it is a significant fraction, we do not need an accurate number, so let’s just pick some reasonable number, say 30%. This would require at least a 10 fold increase in the number of piano teachers. This would be great for students because one of the big problems today is finding good teachers. In any one area, there are presently only a few teachers and the students have little choice. The number of pianos sold would also have to increase, probably by something in excess of 300%. Although many homes already have pianos, many of them are not playable. Since most of the new pianists will be at an advanced level, the number of good grand pianos needed will increase by an even larger percentage.</P>
<P>By using this book as a basic starting point for practice methods, piano teachers can concentrate on what they do best: teaching how to make music. Since this is what teachers have been doing all along, there will be only minor new changes needed in how teachers teach. The only new element is the addition of practice methods that take very little time to learn. The biggest change, of course, is that teachers will be liberated from the old slow process of teaching technique. It will be much easier for teachers to decide what to teach because technical difficulties will be much less of an impediment. Within a few teacher/student generations, the quality of teachers will improve dramatically which will further accelerate the learning rates of future students.</P>
<P>Is an increase of 100 times in the population of pianists reasonable? What would they do? They certainly can’t all be concert pianists and piano teachers. The very nature of how we view piano playing will change. First of all, the piano will, by then, become a standard second instrument for all musicians, because it will be so easy to learn and there will be pianos everywhere. The joy of playing piano will be enough reward for many. The zillions of music lovers who could only listen to recordings can now play their own music -- a much more satisfying experience. As anyone who has become an accomplished pianist will tell you, once you get to that level, you cannot help but compose music. Thus a piano revolution will also ignite a revolution in composition, and new compositions will be in great demand because many pianists will not be satisfied with playing “the same old things”. Pianists will be composing music for every instrument because of the development of keyboards with powerful software and every pianist will have an acoustic piano and an electronic keyboard, or a dual instrument (see below). The large supply of good keyboardists would mean that entire orchestras will be created using keyboard players. Another reason why the piano would become universally popular is that it will be used as a method for increasing the IQ of growing infants. Brain research will certainly reveal that the intelligence can be improved by proper brain stimulation during its early developmental stages. Since there are only two inputs into the infant’s brain, auditory and visual, and the auditory is initially much more advanced than visual, music is the most logical means for influencing the brain during early development.</P>
<P>With such huge forces at work, the piano itself will evolve rapidly. First, the electronic keyboard will increasingly intrude into the piano sector. The shortcomings of the electronic pianos will continue to decrease until the electronics become indistinguishable from the acoustics. Regardless of which instrument is used, the technical requirements will be the same. By then, the acoustic pianos will have many of the features of the electronics: they will be in tune all the time (instead of being out of tune 99% of the time, as they are now), you will be able to change temperaments by flicking a switch, and midi capabilities will be easily interfaced with the acoustics. The acoustics will never completely disappear because the art of making music using mechanical devices is so fascinating. In order to thrive in this new environment, piano manufacturers will need to be much more flexible and innovative.</P>
<P>Piano tuners will also need to adapt to these changes. All pianos will be self-tuning, so income from tuning will decrease. However, pianos in tune 100% of the time will need to be voiced more frequently, and how hammers are made and voiced will need to change. It is not that today’s pianos do not need voicing just as much, but when the strings are in perfect tune, any deterioration of the hammer becomes the limiting factor to sound quality. Piano tuners will finally be able to properly regulate and voice pianos instead of just tuning them; they can concentrate on the quality of the piano sound, instead of just getting rid of dissonances. Since the new generation of more accomplished pianists will be aurally more sophisticated, they will demand better sound. The greatly increased number of pianos and their constant use will require an army of new piano technicians to regulate and repair them. Piano tuners will also be much more involved in adding and maintaining electronic (midi, etc.) capabilities to acoustics. Therefore, the piano tuners’ business will extend into the maintenance and upgrading of electronic pianos. Thus most people will either have a hybrid or both an acoustic and electronic piano.</P>
<LI><P><B>The Future of Education</B></P></LI>
<P>The Internet is obviously changing the nature education. One of my objectives in writing this book on the WWW is to explore the possibilities of making education much more cost effective than it has been. Looking back to my primary education and college days, I marvel at the efficiency of the educational processes that I had gone through. Yet the promise of much greater efficiency via the internet is staggering by comparison. My experience thus far has been very educational. Here are some of the advantages of internet based education:<BR>
(i) No more waiting for school buses, or running from class to class; in fact no more cost of school buildings and associated facilities.<BR>
(ii) No costly textbooks. All books are up-to-date, compared to many textbooks used in universities that are over 10 years old. Cross referencing, indexing, topic searches, etc., can be done electronically. Any book is available anywhere, as long as you have a computer and internet connection.<BR>
(iii) Many people can collaborate on a single book, and the job of translating into other languages becomes very efficient, especially if a good translation software is used to assist the translators.<BR>
(iv) Questions and suggestions can be emailed and the teacher has ample time to consider a detailed answer and these interactions can be emailed to anyone who is interested; these interactions can be stored for future use.<BR>
(v) The teaching profession will change drastically. On the one hand, there will be more one-on-one interactions by email, video-conferencing, and exchange of data (such as audio from a piano student to the teacher) but on the other, there will be fewer group interactions where the group of students physically assembles in one classroom. Any teacher can interact with the “master text book center” to propose improvements that can be incorporated into the system. And students can access many different teachers, even for the same topic.<BR>
(vi) Such a system would imply that an expert in the field cannot get rich writing the best textbook in the world. However, this is as it should be -- education must be available to everyone at the lowest cost. Thus when educational costs decrease, institutions that made money the old way must change and adopt the new efficiencies. Wouldn’t this discourage experts from writing textbooks? Yes, but you need only one such “volunteer” for the entire world; in addition, the internet has already spawned enough such free systems as Linux, browsers, Adobe Acrobat, etc., that this trend is not only irreversible but well established. In other words, the desire to contribute to society becomes a large factor in contributing to education. For projects that provide substantial benefits to society, funding mechanisms (government, philanthropists, and sponsoring businesses) will certainly evolve.<BR>
(vii) This new paradigm of contributing to society may bring about even more profound changes to society. One way of looking at business as conducted today is that it is highway robbery. You charge as much as you can regardless of how much or how little good your product does to the buyer. In an accurate accounting paradigm, the buyer should always get his money’s worth. That is the only situation in which that business can be viable in the long run. This works both ways; well-run businesses should not be allowed to go bankrupt simply because of excessive competition. In an open society in which all relevant information is immediately available, we can have financial accounting that can make pricing appropriate to the service. The philosophy here is that a society consisting of members committed to helping each other succeed will function much better than one consisting of robbers stealing from each other. In particular, in the future, practically all basic education should be essentially free. This does not mean that teachers will lose their jobs because teachers can greatly accelerate the learning rate and should be paid accordingly.</P>
<P>It is clear from the above considerations that free exchange of information will transform the educational (as well as practically every other) field. This book is one of the attempts at taking full advantage of these new capabilities.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140797527994188802006-02-24T08:11:00.000-08:002006-02-24T08:12:08.006-08:00Learning Rate Calculation<P>Here is my crude attempt to mathematically calculate the piano learning rate of the methods of this book. The result indicates that it is about 1000 times faster than the intuitive method. The huge multiple of 1000 makes it unnecessary to calculate an accurate number in order to show that there is a big difference. This result appears plausible in view of the fact that many students who worked hard all their lives using the intuitive method are not able to perform anything significant, whereas a fortunate student who used the correct learning methods can become a concert pianist in less than 10 years. It is clear that the difference in practice methods can make the difference between a lifetime of frustration and a rewarding career in piano. Now, “1000 times faster” does not mean that you can become a pianist in a millisecond; all it means is that the intuitive methods are 1000 times <I>slower</I> than the good methods. The conclusion we should draw here is that, with the proper methods, our learning rates should be pretty close to those of the famous composers such as Mozart, Beethoven, Liszt, and Chopin. Remember that we have certain advantages not enjoyed by those past geniuses. They did not have those wonderful Beethoven sonatas, Liszt and Chopin etudes, etc., with which to acquire technique, or those Mozart compositions with which to benefit from the “Mozart effect”, or books like this one with an organized list of practice methods. Moreover, there are now hundreds of time-proven methods for using those compositions for acquiring technique (Beethoven often had difficulty playing his own compositions because nobody knew the correct or wrong way to practice them). An intriguing historical aside here is that the only common material available for practice for all of these great pianists was Bach’s compositions. Thus, we are led to the idea that studying Bach may be sufficient for acquiring most basic keyboard skills.</P>
<P>I will try to make a detailed calculation starting with the most fundamental precepts and progressing to the final result without jumping over unknown gaps. In this way, if there are errors in this calculation, it can be refined as we improve our understanding of how we acquire technique. This is, obviously, the scientific approach. There is nothing new in these calculations except for their application to musical learning. The mathematical material is simply a review of established algebra and calculus.</P>
<P>Mathematics can be used to solve problems in the following way. First, you define the conditions that determine the nature of the problem. If these conditions have been correctly determined, they allow you to set up what are called differential equations; these are accurate, mathematical statements of the conditions. Once the differentials equations are set up, mathematics provides methods for solving them to provide a function which describes the answers to the problems in terms of parameters that determine these answers. The solutions to the problems can then be calculated by inserting the appropriate parameter values into the function.</P>
<P>The physical principle we use to derive our learning equation is the linearity with time. Such an abstract concept may seem to have nothing to do with piano and is certainly non-biological, but it turns out that, that is exactly what we need. So let me explain the concept of “the linearity with time”. It simply means proportional to time. For example, if we learn an amount of technique L (stands for Learning) in time T, then if we repeat this process again a few days later, we should learn another increment L in the same T. Thus we say that L is linear with respect to T in the sense that they are proportional; in 2T, we should learn 2L. Of course, we know that learning is highly non-linear. If we practice the same short segment for 4 hours, we are likely to gain a lot more during the first 30 minutes than during the last 30 minutes. However, we are talking about an optimized practice session averaged over many practice sessions that are conducted over time intervals of years (in an optimized practice session, we are not going to practice the same 4 notes for 4 hours!). If we average over all of these learning processes, they tend to be quite linear. Certainly within a factor of 2 or 3, linearity is a good approximation, and that amount of accuracy is all we need. Note that linearity does not depend, to first approximation, on whether you are a fast learner or a slow learner; this changes only the proportionality constant. Thus we arrive at the first equation:<BR><BR>
L = kT (Eq. 1.1),<BR><BR>
where L is an increment of learning in the time interval T and k is the proportionality constant. What we are trying to find is the time dependence of L, or L(t) where t is time (in contrast to T which is an interval of time). Similarly, L is an increment of learning, but L(t) is a function.</P>
<P>Now comes the first interesting new concept. We have control over L; if we want 2L, we simply practice twice. But that is not the L that we retain because we <I>lose</I> some L over time after we practice. Unfortunately, the more we know, the more we can forget; that is, the amount we forget is proportional to the original amount of knowledge, L(O). Therefore, assuming that we acquired L(O), the amount of L we lose in T is:<BR><BR>
L = -kTL(O) (Eq. 1.2),<BR><BR>
where the k’s in equations 1.1 and 1.2 are different, but we will not re-label them for simplicity. Note that k has a negative sign because we are losing L. Eq. 1.2 leads to the differential equation<BR><BR>
dL(t)/dt = -kL(t) (Eq. 1.3)<BR><BR>
where “d” stands for differential (this is all standard calculus), and the solution to this differential equation is<BR><BR>
L(t) = Ke(expt.-kt) (Eq. 1.4),<BR><BR>
where “e” is a number called the natural logarithm which satisfies Eq. 1.3, and K is a new constant related to k (for simplicity, we have ignored another term in the solution that is unimportant at this stage). Eq. 1.4 tells us that once we learn L, we will immediately start to forget it exponentially with time if the forgetting process is linear with time.</P>
<P>Since the exponent is just a number, k in Eq. 1.4 has the units of 1/time. We shall set k = 1/T(k) where T(k) is called the characteristic time. Here, k refers to a specific learning/forgetting process. When we learn piano, we learn via a myriad of processes, most of which are not well understood. Therefore, determining accurate values for T(k) for each process is generally not possible, so in the numerical calculations, we will have to make some “intelligent guesses”. In piano practice, we must repeat difficult material many times before we can play them well, and we need to assign a number (say, “i”) to each practice repetition. Then Eq. 1.4 becomes<BR><BR>
L(i,t,k) = K(i)e(expt.-t[i]/T[k]) (Eq. 1.5),<BR><BR>
for each repetition i and learning/forgetting process k. Let’s examine some relevant examples. Suppose that you are practicing 4 parallel set notes in succession, playing rapidly and switching hands, etc., for 10 minutes. We assign i = 0 to one parallel set execution, which may take only about half a second. You might repeat this 10 or 100 times during the 10 minute practice session. You have learned L(0) after the first parallel set. But what we need to calculate is the amount of L(0) that we retain after the 10 minute practice session. In fact, because we repeat many times, we must calculate the cumulative learning from all of them. According to Eq. 1.5, this cumulative effect is given by summing the L’s over all the parallel set repetitions:<BR><BR>
L(Total) = Sum(over i)K(i)e(expt.-t[i]/T[k]) (Eq.1.6).<BR><BR></P>
<P>Now let’s put in some numbers into Eq. 1.6 in order to get some answers. Take a passage that you can play slowly, HT, in about 100 seconds (intuitive method). This passage may contain 2 or 3 parallel sets that are difficult and that you can play rapidly in less than a second, so that you can repeat them over 100 times in those 100 seconds (method of this book). Typically, these 2 or 3 difficult spots are the only ones holding you back, so if you can play them well, you can play the entire passage at speed. Of course, even with the intuitive method, you will repeat it many times, but let’s compare the difference in learning for each 100 second repetition. For this quick learning process, our tendency to “lose it” is also fast, so we can pick a “forgetting time constant” of around 30 seconds; that is, every 30 seconds, you end up forgetting almost 30% of what you learned from one repetition. Note that you never forget everything even after a long time because the forgetting process is exponential -- exponential decays never reach absolute zero. Also, you can make many repetitions in a short time for parallel sets, so these learning events can pile up quickly. This forgetting time constant of 30 seconds depends on the mechanism of learning/forgetting, and I have chosen a relatively short one for rapid repetitions; we shall examine a much longer one below.</P>
<P>Assuming one parallel set repetition per second, the learning from the first repetition is e(expt.-100/30) = 0.04 (you have 100 seconds for forget the first repetition), while the last repetition gives e(expt.-1/30) = 0.97, and the average learning is somewhere in between, about 0.4 (we don’t have to be exact, as we shall see). and with over 100 repetitions, we have over 40 units of learning for the use of parallel sets. For the intuitive method, we have a single repetition or e(expt. -100/30) = 0.04. The difference is a factor of 40/0.04 = 1,000! With such a large difference factor, we do not need much accuracy to demonstrate that there is a big difference. The actual difference in learning may be even bigger because the intuitive method repetition is at slow speed whereas the parallel set repetition rate is at, or even faster than, the final speed.</P>
<P>The 30 second time constant used above was for a “fast” learning process, such as that associated with learning <I>during</I> a single practice session. There are many others, such as technique acquisition by PPI (post practice improvement). After any rigorous conditioning, your technique will improve by PPI for a week or more. The rate of forgetting, or technique loss, for such slow processes is not 30 seconds, but much longer, probably several weeks. Therefore, in order to calculate the total difference in learning rates, we must calculate the difference for all known methods of technique acquisition using the corresponding time constant, which can vary considerably from method to method. PPI is largely determined by conditioning, and conditioning is similar to the parallel set repetition calculated above. Thus the difference in PPI should also be about 1,000 times.</P>
<P>Once we calculate the most important rates as described above, we can refine the results by considering other factors that influence the final results. There are factors that make the methods of this book slower (initially, memorizing may take longer than reading, or HS may take longer than HT because you need to learn each passage 3 times instead of once, etc.) and factors that make them faster (such as learning in short segments, getting up to speed quickly, avoiding speed walls, etc.). There are many more factors that make the intuitive method slower, so that the above “1000 times faster” result may be an under-estimate. However, it is probably not possible to take full advantage of the 1000 times factor, since most students may already be using some of the ideas of this book.</P>
<P>The effects of speed walls are difficult to calculate because speed walls are artificial creations of each pianist and I do not know how to write an equation for them. Experience tells us that the intuitive method is susceptible to speed walls. The methods of this book provide many ways of avoiding them. Moreover, speed walls are clearly defined here so that it is possible to pro-actively avoid them during practice. Parallel sets are the most powerful tools for avoiding them because speed walls do not generally form when you decrease speed from high speed. Therefore, speed walls greatly retard the learning rate for intuitive methods. Some teachers who do not understand speed walls adequately will prohibit their students from practicing anything risky and fast, thus slowing progress even more, even when this slow play succeeds in completely avoiding speed walls. When all these factors are taken into account we come to the conclusion that the “up to 1000 times faster” result is basically correct. We also see that the use of parallel sets, practicing difficult sections first, practicing short segments, and getting up to speed quickly, are the main factors that accelerate learning. HS practice, relaxation, and early memorization are some of the tools that enable us to optimize the use of these accelerating methods.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140797330143687712006-02-24T08:08:00.000-08:002006-02-24T08:08:50.146-08:00Thermodynamics of Piano Playing<P>An important field of mathematics is the study of large numbers. Even when single events of a particular type are not predictable, large numbers of such events often behave according to strict laws. Although the energies of individual molecules of water in a glass may differ considerably, the temperature of the water can stay very constant and can be measured with high accuracy. Does piano playing have an analogous situation that would allow us to apply the laws of large numbers and thereby draw some useful conclusions?</P>
<P>Piano playing is a complex process because of a large number of variables that enter into the production of music. The study of large numbers is accomplished by counting the "number of states" of a system. The grand total of meaningful states so counted might be called the "canonical ensemble", a meaningful assembly that sings together a tune that we can decipher. Believe it or not, "canonical ensemble" (see Statistical Mechanics by Kerson Huang, Wily, 1963, P. 75) is legitimate thermodynamic terminology! Therefore, all we need to do is to calculate the canonical ensemble, and when finished, we simply apply the known mathematical rules of large systems (i.e., thermodynamics) and voila! We are done!</P>
<P>The variables in question here are clearly the different motions of the human body, especially those parts important in playing the piano. Our job is to count all the ways in which the body can be moved in playing the piano; this is clearly a very large number; the question is, is it large enough for a meaningful canonical ensemble?</P>
<P>Since no one has ever attempted to calculate this canonical ensemble, we are exploring new territory here, and I will attempt only a very approximate estimate. The beauty of canonical ensembles is that, in the end, if the calculations are correct (a legitimate concern for something this new), the method used to arrive at it is usually immaterial -- you always arrive at the same answers. We calculate the ensemble by listing all the relevant variables, and counting the total parameter space of these variables. So here we go.</P>
<P>Let's start with the fingers. Fingers can move up and down, sideways, and be curled or straight (three variables). Say there are 10 measurably different positions for each variable (parameter space = 10); counting only the number of 10s, we have 4 so far, including the fact that we have 10 fingers. There are actually many more variables (such as rotating the fingers around each finger axis) and more than 10 measurable parameters per variable, but we are counting only those states that can be reasonably used to play the piano, for a given piece of music. The reason for this restriction is that we will be using the results of these calculations for comparing how two persons play the same piece, or how one person would play it twice in a row. This will become clearer later on.</P>
<P>Now the palms can be raised or lowered, flexed sideways, and rotated around the axis of the forearm. That's 3 more 10s for a total of 7. The forearm can be raised or lowered, and swung sideways; new total is 9. The upper arm can be swung forward-backwards, or sideways; new total is 11. The body can be moved forwards-backwards, and sideways: new total is 13. Then there are the variables of force, speed, and acceleration, for a total of at least 16. Thus the total parameter space of a pianist has many more than 10(exp)16 states (one followed by 16 zeros!). The actual number for a given piece of music is many orders of magnitude larger because of the above calculation is only for one note and a typical piece of music contains thousands or tens of thousands of notes. The final parameter space is therefore about 10(exp)20. This is approaching the ensemble space for molecules; for example, one cc of water has 10(exp)23 molecules, each with several degrees of freedom of motion and many possible energy states. Since thermodynamics applies to volumes of water very much smaller than 0.0001 cc, the canonical ensemble of the pianist is pretty close to thermodynamic conditions.</P>
<P>If the canonical ensemble of the pianist is nearly thermodynamic in nature, what conclusion can we draw? The most important result is that any single point in this phase space is totally irrelevant, because the chances of your reproducing this particular point is essentially zero. From this result, we can draw some immediate conclusions:</P>
<B><I><P>First Law of Pianodynamics: no two persons can play the same piece of music in exactly the same way. A corollary to this first law is that the same person, playing the same piece of music twice, will never play it exactly the same way.</P></I></B>
<P>So what? Well, what this means is that the notion that listening to someone play might decrease your creativity by your imitating that artist is not a viable idea, since it is never possible to imitate exactly. It really supports the school of thought which claims that listening to good artists play cannot hurt. Each pianist is a unique artist, and no one will ever reproduce her/is music. The corollary provides a scientific explanation for the difference between listening to a recording (which reproduces a performance exactly) and listening to a live performance, which can never be reproduced (except as a recording).</P>
<B><I><P>Second Law of Pianodynamics: we can never completely control every aspect of how we play a given piece.</P></I></B>
<P>This law is useful for understanding how we can unconsciously pick up bad habits, and how, when we perform, the music takes on its own life and in some ways, goes out of our own control. The powerful laws of pianodynamics take over in these cases and it is useful to know our limitations and to know the sources of our difficulties in order to control them as much as possible. It is a truly humbling thought, to realize that after a long, hard practice we could have picked up any number of bad habits without ever even suspecting it. This may in fact provide the explanation of why it is so beneficial to play slowly on the last run-through during practice. By playing slowly and accurately, you are greatly narrowing the ensemble space, and excluding the "bad" ones that are far away from the "correct" space of motions. If this procedure does indeed eliminate bad habits, and is cumulative from practice session to practice session, then it could produce a huge difference in the rate at which you acquire technique in the long run. </P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140797268296711762006-02-24T08:07:00.000-08:002006-02-24T08:07:48.296-08:00Increasing Speed<P>These results also provide the mathematical basis for explaining the well known trick of alternating fingers when playing the same note many times. One might think at first that using just one finger would be easier and offer more control, but that note can be played repetitively faster by playing parallel using as many fingers as you can for that situation, than playing serially.</P>
<P>The need for parallel play also singles out trills as a particularly difficult challenge to play fast because trills must in general be executed with only two fingers. If you tried to trill with one finger, you will hit a speed wall at, say, speed M; if you trill with two fingers, the speed wall will be at 2M (again, ignoring momentum balance). Does mathematics suggest any other way of attaining even higher speeds? Yes: phase truncation.</P>
<P>What you can do is to lower the finger to play the note but then raise the finger only sufficiently to reset the repetition mechanism, before playing the next note. You may need to raise the finger by only 90 degrees instead of the normal 180 degrees. This is what I mean by phase truncation; the unnecessary part of the total phase is truncated off. If the original amplitude of finger travel for the 360 degree motion was 2 cm, with a 180 degree truncation, the finger now moves only 1 cm. This 1 cm can be further reduced until the limit at which the repetition mechanism stops working, at about 5 mm. Phase truncation is the mathematical basis for the fast repetition of the grand and explains why the rapid repetition is designed to work with a short return distance.</P>
<P>A good analogy to gaining speed in this way is the dribbling action of a basketball, as contrasted to the swinging action of a pendulum. A pendulum has a fixed frequency of swing regardless of the swing amplitude. A basketball, however, will dribble faster as you dribble closer to the ground (as you reduce the dribble amplitude). A basketball player will generally have a hard time dribbling until s/he learns this change in dribble frequency with dribble height. A piano acts more like a basketball than a pendulum (fortunately!), and the trill frequency increases with decreasing amplitude until you reach the limit of the repetition mechanism. Note that even with the fastest trill, the backcheck is engaged for a correct trill, because the keys must always be depressed completely. The trill is possible because the mechanical response of the backcheck is faster than the fastest speed that the finger can achieve. </P>
<P>The trill speed is not limited by the piano mechanism except for the height at which the repetition stops working. Thus it is more difficult to trill rapidly with most uprights because phase truncation is not as effective. These mathematical conclusions are consistent with the fact that to trill fast, we need to keep the fingers on the keys and to reduce the motions to the minimum necessary for the repetition mechanism to work. The fingers must press "deeply into the piano" and just lifted sufficiently to activate the repetition mechanism. Furthermore, it helps to use the strings to bounce the hammer back, just as you bounce the basketball off the floor. Note that a basketball will dribble faster, for a given amplitude, if you press down harder on it. On the piano, this is accomplished by pressing the fingers firmly down on the keys and not letting them "float up" as you trill.</P>
<P>Another important factor is the functional dependence of finger motion (purely trigonometric, or hyperbolic, etc.) for controlling tone, staccato, and other properties of the piano sound relating to expression. With simple electronic instruments, it is an easy task to measure the exact finger motion, complete with key speed, acceleration, etc. These characteristics of each pianist's playing can be analyzed mathematically to yield characteristic electronic signatures that can be identified with what we hear aurally, such as angry, pleasant, boisterous, deep, shallow, etc. For example, the motion of the key travel can be analyzed using FFT (fast Fourier transform), and it should be possible, from the results, to identify those motion elements that produce the corresponding aural characteristics. Then, working backwards from these characteristics, it should be possible to decipher how to play in order to produce those effects. This is a whole new area of piano play that has not been exploited yet. This kind of analysis is not possible by just listening to a recording of a famous artist, and may be the most important topic for future research.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0tag:blogger.com,1999:blog-22149718.post-1140797187280724692006-02-24T08:04:00.000-08:002006-02-24T08:06:27.283-08:00Speed Walls<P>Let's assume that a person starts practicing a piece of music by first playing slowly, using mostly serial play, since that is the easiest way (let's ignore chords for the time being). As the finger speed is gradually increased, s/he will naturally hit a speed wall because human fingers can move only so fast. Thus we have mathematically discovered one speed wall, and that is the speed wall of serial play. How do we break this speed wall? We need to find a play method that has no speed limit. That is parallel play. In parallel play, you increase the speed by decreasing the phase difference. That is, speed is numerically proportional to the inverse of the phase difference. Since we know that the phase difference can be decreased to zero (which gives you a chord), we know that parallel play has the potential give you infinite speed and therefore it has no theoretical speed limit. We have arrived at a mathematical foundation for the chord attack!</P>
<P>The distinction between serial and parallel play is somewhat artificial and oversimplified. In reality, practically everything is played parallel. Thus the above discussion served only as an illustration of how to define or identify a speed wall. The actual situation with each individual is too complex to describe (because speed walls are caused by bad habits, stress and HT play), but it is clear that wrong playing methods are what create speed walls and each person has her/is own mistakes that result in speed walls. This is demonstrated by the use of parallel set exercises which overcome the speed walls. This means that speed walls are not always there by themselves, but are <I>created</I> by the individual. Therefore, there is any number of possible speed walls depending on each individual and every individual has a different set of speed walls. There are, of course common classes of speed walls, such as those created by stress, by wrong fingering, by lack of HS technique, lack of HT coordination, etc. It would be, in my opinion, very counterproductive to say that such complex concepts will never be scientifically or mathematically treated. We have to. For example, in parallel play, phase plays a very important part. By decreasing the phase to zero, we can play infinitely fast, in principle.</P>
<P>Can we really play infinitely fast? Of course not. So then what is the ultimate parallel speed limit, and what mechanism creates this limit? We know that different individuals have different speed limits, so the answer must include a parameter that depends on the individual. Knowing this parameter will tell us how to play faster! Clearly, the fastest speed is determined by the smallest phase difference that the individual can control. If the phase difference is so small that it cannot be controlled, then "parallel play speed" loses its meaning. How do we measure this minute phase difference for each individual? This can be accomplished by listening to her/is chords. The accuracy of chord play (how accurately all the notes of the chord can be played simultaneously) is a good measure of an individual's ability to control the smallest phase differences. Therefore, in order to be able to play parallel fast, you must be able to play accurate chords. This means that, when applying the chord attack, you must first be able to play accurate chords before proceeding to the next step.</P>
<P>It is clear that there are many more speed walls and the particular speed wall and the methods for scaling each wall will depend on the type of finger or hand motion. For example you can attain infinite speed with parallel play only if you have an infinite number of fingers (say, for a long run). Unfortunately, we have only ten fingers and often only five are available for a particular passage because the other five are needed to play other parts of the music. As a rough approximation, if serial play allows you to play at a maximum speed of M, then you can play at 2M using two fingers, 3M using three fingers, etc., serially. The maximum speed is limited by how rapidly you can recycle these fingers. Actually, this is not quite true because of momentum balance (it allows you to play faster), which will be treated separately below. Thus each number of available fingers will give you a different new speed wall. We therefore arrive at two more useful results. (1) there can be any number of speed walls, and (2) you can change your speed wall by changing your fingering; in general, the more fingers you can use in parallel before you need to recycle them, the faster you can play. Putting it differently, most conjunctions bring with them their own speed walls.</P>Rosyhttp://www.blogger.com/profile/10835542615317388768noreply@blogger.com0